LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 

Class 


ENERGY 


OF  THE 

UNIVERSITY 

OF 


Published   by  the 

McGraw-Hill    Book.  Company 


to  the  BookDepartments  of  tKe 

McGraw  Publishing  Company  Hill  Publishing  Company 

r\ibliahers  of  Books  "for 

Elec  trical  World  The  Engineering  and  Mining  Journal 

TKe  Engineering  Record  Power  and  The  Engineer 

Electric  Railway  Journal  American   Machinist 


ENERGY 


WORK,  HEAT  AND 
TRANSFORMATIONS 


BY 


SIDNEY  A.  REEVE,  M.E. 


OF  THE 

UNIVERSITY 

OF 


NEW  YORK 

McGRAW-HILL   BOOK   COMPANY 
1909 


Copyright,  1909,  by  the  McGRAW  HILL  BOOK  COMPANY 


PREFACE 

The  earlier  chapters  of  this  work  are  self-explanatory.  The 
later  ones  justify  some  discussion  of  point  of  view. 

The  writer  is  not  a  physicist.  Educated  and  trained  as  an 
engineer,  his  call  to  the  teacher's  chair  led  him  to  arrange  his 
views  as  to  natural  principles  with  greater  care  than  is  common 
with  engineers.  The  ideas  promulgated  in  the  following  pages 
are  in  answer  to  questions  received  from  bright-minded  students 
a  dozen  years  ago — questions  which  were  sensibly  asked,  but 
which  "stumped"  the  teacher  for  years  for  an  adequate,  equally 
sensible  reply. 

His  efforts  at  the  comprehension  of  thermodynamic  action 
have  led  him  to  trespass,  perhaps,  upon  the  domain  of  the 
physicist.  For  the  discussion  of  matters  so  intricate  as  molecular 
dynamics  a  thorough  familiarity  with  experimental  and  mathe- 
matical physics  might  seem  indispensable.  No  one  can  regret 
more  than  the  writer  his  lack  of  this.  His  apologies  for  the  conse- 
quent shortcomings  of  this  little  book  are  profuse  in  proportion. 

Yet,  should  this  situation  arouse  criticism  or  doubt,  the 
answer  is  easy.  Why  have  not  the  professional  physicists  long 
ago  done  this  same  thing,  that  it  might  have  been  done  far 
better?  The  materials,  opportunity  and  demand  have  long 
existed.  Whatever  question  may  arise  as  to  the  significance,  or 
even  the  definition,  of  the  more  recent  data  of  experimental 
physics,  there  can  be  none  as  to  the  long  established  principles 
of  celestial  mechanics.  There  is  as  little  as  to  the  only  less 
venerable  data  as  to  thermal  processes.  For  nearly  a  century 
there  has  been  virtually  no  question  as  to  the  mechanical  nature 
of  heat.  Yet  these  things  cannot  be  accepted  by  any  teacher  of 
thermodynamics  without  enforcing  conclusions  substantially  dif- 
ferent from  those  now  commonly  taught.  Therefore  the  writer 
cannot  regard  the  concepts  set  forth  in  this  little  book  as  aught 
else  than  the  indispensable  premises  for,  rather  than  the  remote 
conclusions  from,  experimental  and  mathematical  physics. 

The  writer  would  never  wilfully  question  the  doctrine  that 
accurate  data  are  essential  to  progress.  Scientific  concepts  cannot 


209614 


advance  without  them.  But  what  has  been  forgotten  is  that  they 
are  a  means  to  an  end,  not  an  end  in  themselves ;  and  that  end  is 
the  better  understanding  of  nature's  ways.  Science  serves 
humanity  only  as  it  substitutes  scientific  concepts  for  supersti- 
tious guess-work.  The  accumulation  of  unending  columns  of 
figures  and  physical  constants,  even  with  religious  accuracy,  is  as 
futile  for  the  uplift  of  the  race  as  is  the  accumulation  of  vast 
hoards  of  dollars,  however  conscientiously  accounted  to  the  last 
penny.  Each  may  be  turned  to  humanitarian  ends.  But  until 
it  is,  like  a  prostrate  ladder,  it  constitutes  a  trap  for  the  feet  of 
the  unwary,  rather  than  a  pathway  erected  to  higher  things. 

The  book,  therefore,  is  merely  an  attempt  to  fit  together 
(i)  the  Newtonian  mechanics,  (2)  the  doctrine  that  heat  is 
mode  of  motion,  and  (3)  the  dozen  or  so  well  known  facts  of 
thermal  action,  into  a  consistent  whole  which  may  serve  as  an 
engineer's  idea  of  heat  and  heat-action.  It  was  originally  pre- 
pared for  publication  in  the  periodical  press,  and  some  of  the 
earlier  portions  appeared,  in  preliminary  form,  in  the  columns 
of  The  Engineer  (London).  Some  traces  of  this  genesis  may 
be  noticed  in  the  pages  of  the  book. 

The  writer  wishes  to  acknowledge  his  indebtedness  to  the 
Sheffield  Scientific  School,  of  Yale  University,  for  helpful 
facilities  for  work. 

NEW  HAVEN,  CONN.,  July,  1909. 


TABLE  OF  CONTENTS 

PAGE 

CHAPTER  I.           Mechanical    Energy 7 

CHAPTER  II.          Free  and  Vibratory  Energies    18 

CHAPTER  III.        The  Mean  Energetic  Condition  and  the  Energy-fund  32 

CHAPTER  IV.        The  Two  Factors  or  Dimensions  of  Energy 48 

CHAPTER  V.          The  Extreme  or  Critical  Energetic  Conditions 61 

CHAPTER  VI.        The  General  Nature  of  Mechanical  Energy    78 

CHAPTER  VII.      What  is   Heat?    89 

CHAPTER  VIII.     The   Thermal   Diagram    95 

CHAPTER  IX.        Mechanical  Concepts  of  Thermal   Phenomena    106 

A.  Pressure  and  Volume. 

CHAPTER  X.          Mechanical     Concepts    of     Pressure     and    Volume 

(cont.)     117 

CHAPTER  XI.        The  Two  Basic  Thermal   Processes:   Heat-transfer 

and    Work-performance    129 

CHAPTER  XII.       Mechanical  Concepts  of  Thermal  Phenomena     141 

B.  Temperature  and  Entropy. 

CHAPTER  XIII.     The    Energetic    Cycle     158 

CHAPTER  XIV.     Reversed  and  Irregular  Cycles    175 

CHAPTER  XV.       Thermal    Equilibrium    182 

CHAPTER  XVI.     Transformations    and   Conservations     208 


ENERGY 

CHAPTER  I. 

MECHANICAL  ENERGY. 

The  muscular  system  of  our  modern  body  politic  is  its  array 
of  energy-producing  machines.  Man  has  magnified  his  own  tiny 
energies  with  power  borrowed  from  nature.  His  land-carriers 
put  to  scorn  the  elephant ;  his  ships  make  pygmies  of  the  whales. 
Take  from  mankind  to-morrow  this  artificial  multiplication  of 
its  abilities  to  "overcome  resistance,"  and  within  a  month  its 
ranks  will  be  decimated  by  starvation.  Within  a  decade  it  will 
have  become  a  mob  of  howling  beasts,  fighting  for  the  insufficient 
means  of  existence — slipped  back  centuries  in  growth  of  civiliza- 
tion as  well  as  of  population. 

Yet  to-day,  in  spite  of  this  overwhelming  importance  of 
energy  in  modern  community-life,  there  exists  no  idea  of  its 
nature  more  exact  than  the  general  one  that  it  is  an  aid  in  the 
overcoming  of  resistance.  Force  times  space,  or  mass  under- 
going acceleration,  equals  energy.  Thus  far  we  appear  to  pro- 
ceed coherently.  But  it  is  not  far  enough,  when  the  chief 
business  of  an  increasing  fraction  of  the  human  race  is  the 
transformation  and  transportation  of  energy — mechanical,  hy- 
draulic, electrical,  chemical,  thermal,  etc. ;  and  even  in  going  that 
far  we  quite  lack,  it  appears,  a  complete  agreement  between  the 
authorities. 

Then,  again,  the  bulk  of  all  this  current  supply  of  energy  to 
the  human  race  is  obtained  in  the  form  of  heat.  Yet  as  to  what 
heat  is  we  know  virtually  nothing.  It  is  commonly  described,  in 
an  attempt  to  explain  its  nature,  as  a  "mode  of  motion"  be- 
tween the  particles  of  the  hot  body.  But  an  explanation  of  an 
obscure  thing,  in  order  to  help  rather  than  hinder  the  under- 
standing, must  speak  in  terms  of  familiar  things ;  and  when  this 
"explanation"  of  heat  in  terms  of  mass  and  motion  is  once 
examined,  it  is  found  to  contain  two  elements  which  are  even 
more  unfamiliar  and  obscure  than  is  heat  itself. 

7 


8  ENERGY 

The  first  of  these  elements  is  that  "perfect  elasticity"  which 
must  be  attributed  to  the  constantly  rebounding  particles  upon 
the  molecular  theory  of  heat.  For,  since  heat  continues  indefi- 
nitely in  a  hot  body,  so  long  as  none  is  abstracted,  each  rebound 
must  occur  millions  of  times  per  second,  with  perfect  elasticity, 
in  order  that  the  continuous  existence  of  heat  may  be  explained 
mechanically. 

Yet  such  a  thing  as  a  rebound,  after  collision,  with  perfect 
elasticity,  is  unknown  in  nature.  In  fact,  it  is  one  of  the 
fundamental  doctrines  of  thermal  science  that  motion  cannot 
occur  in  nature  without  some  loss  of  energy  inelastically,  in 
either  friction  or  impact.  If  there  be  in  nature  such  a  thing  as 
perfect  elasticity,  human  observation  has  never  yet  discerned  it. 
Therefore,  the  "mode  of  motion"  explanation  of  heat  is  an 
explanation  of  one  thing,  namely,  heat,  which,  albeit  mystical 
in  its  nature,  is  yet  familiar  to  every  schoolboy,  in  terms  of 
another  thing,  namely,  perfect  elasticity,  which  no  one  has  ever 
yet  known  to  exist  anywhere.  The  only  thing  plain  about  such 
an  explanation  is  that  it  doesn't  explain. 

The  other  element  of  common  sense  which  is  wanting  in  this 
alleged  explanation  of  heat  is  an  exact  and  familiar  idea  of 
mechanical  energy  itself.  It  is  of  no  use  to  explain  heat  as  a 
microscopic  form  of  mechanical  energy  if  we  do  not  know  what 
mechanical  energy  is.  The  term  mechanical  energy,  it  is  true,  is 
used  most  familiarly  by  engineers;  yet,  astounding  as  the  fact 
may  seem,  it  is  nevertheless  true  that  there  exists  to-day  no 
agreement  between  the  authorities  as  to  just  what  mechanical 
energy  is.  The  statements  concerning  it  which  are  rife  among 
the  engineers,  the  teachers  of  engineering  and  their  text-books, 
can  all  be  reduced  quickly  to  either  absurdities  or  approxima- 
tions. To  "explain,"  therefore,  that  heat  is  a  "mode  of  motion," 
when  not  one  student  of  engineering  in  a  thousand  has  ever 
been  taught  what  modes  of  motion  are  possible  in  nature  and 
what  are  impossible,  is  also  a  procedure  of  doubtful  utility. 

Nevertheless,  there  is  no  escape  from  the  overwhelming  and 
rapidly  accumulating  evidence  that  heat  is  capable  of  a  real  and 
true  explanation,  as  an  intricate  form  of  mechanical  energy. 
But  progress  can  be  made  in  this  idea  only  by  a  careful  in- 
vestigation, first,  of  what  mechanical  energy  itself  truly  is,  in 
nature;  and,  secondly,  of  what  conclusions  should  be  pointed 


MECHANICAL  ENERGY  9 

therefrom  as  to  the  nature  of  heat,  in  natural  common  sense. 
Fortunately,  every  student  of  engineering  science  is  equipped 
for  both  of  these  tasks,  although  few  may  have  been  led  to 
perform  them.  A  knowledge  of  Kepler's  and  Newton's  laws  of 
motion  and  force,  a  little  analytical  geometry,  and  enough  of  the 
concepts  of  the  calculus  to  permit  thinking  of  millions  of  mole- 
cules at  once,  without  becoming  confused  as  to  what  they  may 
and  may  not  do — this  is  all  that  is  required. 

Mechanical  Energy.  Mechanical  force — that  is  to  say, 
force  manifested  in  such  a  way  that  the  human  understanding 
can  follow  its  origin,  dimensions  and  destination — is  found,  in 
nature,  exerted  through  space,  in  only  two  elementary  ways. 
The  first  is  when  gravitation,  which  acts  at  all  times  between 
each  two  portions  of  matter  in  the  universe  to  hold  them 
together,  finds  chance  to  move  them  and  to  destroy  their  relative 
separation.  Such  action  is  called  a  manifestation  of  "potential" 
or  "positional"  or  "space"  energy,  visible  most  familiarly  in  the 
falling  of  weights.  The  other  instance  occurs  when  either  of  the 
forces  which  everywhere  and  at  all  times  tend  to  hold  each  two 
bodies  in  the  universe  both  together  and  apart  finds  chance  to 
produce  or  destroy  their  relative  motion.  Such  action  is  called  a 
manifestation  of  "kinetic"  or  "accelerative"  or  "motion"  energy. 
It  is  visible  in  pure  mechanics  only  in  the  action  of  centrifugal 
force. 

It  is  to  be  noted  carefully — much  more  carefully  than  the 
text-books  require — that  it  is  neither  space  nor  motion  alone 
which  constitutes  energy,  but  change  in  space  or  motion.  A 
suspended  weight  possesses  no  energy  if  it  never  can  fall.  A 
flying  cannon-ball  can  overcome  no  resistance  if  nothing  ever 
interferes  with  it  to  stop  it. 

But  there  is  another  fact  which  is  of  even  greater  importance 
in  these  definitions,  and  which  the  text-books  quite  omit  alto- 
gether. This  is  that  in  any  energetic  manifestation  there  are 
always  involved  at  least  two  bodies.  No  single  body  may  ever 
possess  energy.  When  it  is  remembered  that  all  of  the  com- 
monly taught  mathematical  expressions  for  energy  include  but  a 
single  symbol  for  mass,  and  that  in  general  terms  energy  is  uni- 
versally described  as  an  attribute  of  mass,  this  statement  cannot 
be  too  strongly  emphasized.  Indeed,  the  only  proper  way  to 
state  this  idea  is  to  say  that  energy  can  exist  only  where  mass 


10  ENERGY 

js  not — namely,  between  two  mass-portions.  It  is  just  as  broad 
a  truth  to  say  that  mas?  and  energy  cannot  occupy  the  same 
place  at  the  same  time  as  it  is  to  say  that  two  portions  of  matter 
cannot  do  so. 

To  explain  more  in  detail:  A  ton  of  water  in  a  mill-pond, 
we  say,  possesses  energy.  That  is,  it  will  overcome  resistance  in 
its  fall  toward  the  tail-race  But  why  will  it  fall,  and  whither? 
What  gives  it  its  weight  and  energy  1 

Its  separation  from  the  earth  gives  it  its  energy.  So  soon  as 
it  succeeds  in  reuniting  itself  with  the  earth  its  energy  is  gone. 
The  energy  cannot  be  said  to  belong  to  the  water,  for  without 
the  propinquity  of  the  earth  the  water  would  have  no  energy.  It 
cannot  be  said  to  belong  to  the  earth,  for  all  the  earth's  gigantic 
mass  would  be  inert  and  helpless  to  perform  work  were  it  a  truly 
solid  unit,  with  no  fragments  split  off,  like  the  water,  to  help  it 
embody  energy.  And,  finally,  the  energy  cannot  be  said  even  to 
belong  to  the  earth  and  water  together,  because  both  earth  and 
water  might  be  present — as  is  the  case,  indeed,  in  the  great 
oceans — without  embodying  any  potential  hydraulic  energy,  be- 
cause there  exists  no  separation  of  earth  and  water,  vertically; 
that  is,  there  is  no  head.  It  is  in  the  special  relationship  existent 
between  the  earth  and  mill-pond  water  that  the  energy  lies,  and 
not  in  either  one,  nor  in  both  together.  Literally,  it  lies  in  both 
apart. 

The  same  statements  hold  true  of  kinetic  energy.  There  is  no 
energy  in  a  flying  cannon-ball,  for  instance,  unless  there  be 
something  to  stop  it.  It  is  only  in  its  stoppage  that  the  projectile 
can  overcome  resistance;  and  it  is  only  a  second  mass-portion 
which  can  do  any  stopping.  No  mere  geometric  point,  nor  co- 
ordinate axis,  can  arrest  a  moving  mass.  To  refer  the  motion 
of  any  mass  to  such  a  geometric  base,  as  a  measure  of  its  energy, 
is  an  error  so  fundamental  that  it  has  adulterated  our  entire 
science  of  energetics.  There  is  only  one  base  of  reference  for 
the  energy  of  any  mass-pair,  and  that  is  its  common  center  of 
mass. 

Kinetic  energy,  therefore,  also  consists  of  a  relationship  be- 
tween two  masses.  It  belongs  to  neither  mass  alone,  nor  even  to 
their  aggregation  alone,  but  to  their  aggregation  when  subdivided 
into  separate  portions  by  relative  motion  between  them. 

In  mechanical   engineering  these  facts  have  long  been  lost 


MECHANICAL  ENERGY  11 

sight  of  (although  long  known),  for  two  reasons.  In  the  first 
place,  one  of  the  two  mass-portions,  in  engineering  problems,  is 
always  the  earth ;  and  the  earth  possesses  so  gigantic  a  mass,  in 
proportion  to  that  of  any  body  of  engineering  magnitude,  that  it 
supplies  an  apparently  fixed  base  of  reference.  Moreover,  its 
great  mass  remains  always  constant,  so  far  as  engineering  instru- 
ments can  perceive,  thus  leaving  the  much  smaller  masses  of  our 
cannon-balls,  railroad  trains,  etc.,  as  the  only  variables. 

In  molecular  mechanics,  however,  it  is  not  only  quite  possible, 
it  is  altogether  probable,  that  the  various  mass-portions  are  of 
the  same  order  of  magnitude;  or,  at  least,  if  there  be  all  classes 
of  magnitudes,  that  there  are  many  of  each  class.  Under  such 
conditions  it  is  of  prime  importance  to  remember  that  there  are 
two  mass-members,  at  least,  in  each  unit  of  energy,  and  that  one 
of  them  is  just  as  likely  to  be  a  variable  as  the  other. 

The  second  way  in  which  the  special  conditions  prevailing  in 
engineering  practice  have  warped  its  concepts  of  energy  away 
from  the  truth  is  seen  in  the  common  idea  of  potential  energy  as 
synonymous  with  "up  and  down."  To  a  student  of  purely  mun- 
dane forces  this  is,  of  course,  natural ;  but  when  one  has  studied 
true  mechanics  long  enough  to  get  away  from  the  special  condi- 
tions of  the  earth's  surface,  it  is  appreciated  that  potential 
energy  refers  only  to  "together  and  apart."  There  is  no  true 
"up"  nor  "down"  in  nature.  It  were  well,  for  instance,  as  we 
stand  by  the  brink  of  a  Niagara,  awed  by  the  thunder  of  its 
action  and  marveling  at  even  that  slight  modicum  of  its  energy 
which  the  power-houses  succeed  in  catching,  to  remember  that 
almost  beneath  our  feet  the  even  more  stupendous  falls  of  the 
Zambesi,  of  more  than  twice  the  height  and  certainly  equal 
power  of  Niagara,  are  thundering  their  watery  masses  in  ex- 
actly the  opposite  direction  from  those  before  our  eyes. 

Yet  in  either  case  the  action  is  the  same.  The  energy  lies  not 
in  the  water  of  either  cataract ;  it  lies  in  the  relationship  existent 
between  water  and  earth.  And  of  this  relationship  no  concept 
can  be  had  by  speaking  of  "up  and  down."  It  is  the  relative 
separation  of  earth  and  water,  above  the  falls  of  either  river, 
which  constitutes  the  energy;  and  the  only  adverbs  which  will 
cover  this  idea  are  "together"  and  "apart."  All  portions  of 
matter  in  the  universe  tend,  by  gravitation,  to  fall  together,  or  to 
consolidate.  All  of  them  also  tend,  by  centrifugal  action,  to  fall 


12  ENERGY 

apart,  or  separate,  or  disintegrate.  Sometimes,  as  on  this  earth's 
surface,  the  former  tendency  is  overwhelmingly  greater  than  the 
latter,  and  the  latter  is  therefore  lost  sight  of.  But  in  many 
other  places  it  is  the  centrifugal  tendency  which  overwhelms 
and  obscures  the  centripetal,  so  that  the  latter  is  forgotten. 
Such  is  usually  the  case  in  the  so-called  permanent  gases,  which 
tend  always  to  expand  indefinitely. 

But  for  our  present  purpose,  viz. :  A  true  statement  of  prin- 
ciples such  as  may  fit  all  cases,  all  that  is  necessary  is  to  keep 
both  facts  constantly  in  mind:  that  at  all  times  and  places  in 
nature  both  tendencies,  the  congregative  and  the  disgregative, 
are  at  work,  in  opposition  to  each  other — sometimes  one  and 
sometimes  the  other  prevailing. 

One  of  our  national  American  humorists  of  a  generation  ago, 
in  promulgating  the  manifold  attractions  of  his  world-famed 
"show,"  advertised  the  exhibition  of  one  of  the  Siamese  twins — 
"the  only  one  which  had  ever  been  successfully  separated."  Not 
that  the  one  had  died,  but  that  it  had  not  yet  been  sepa- 
rated. The  absurd  humor  of  the  remark  never  needed  an  ex- 
planation. And  yet  the  two  great  twin  forces  of  nature — the 
centripetal,  or  gravitational,  and  the  centrifugal;  the  congre- 
gative and  the  disgregative — have  ever  been  solemnly  presented 
to  our  students  of  engineering,  at  different  and  disconnected  times 
and  places,  as  if  each  were  the  only  one  which  had  ever  been 
separated  from  the  other.  They  are  taught  as  if  each  were  an 
independent  natural  phenomenon. 

As  a  matter  of 'fact,  the  action  of  gravitation  is  always  and 
insistently  to  combine  and  unify  mass;  and  if  it  had  its  way 
unhindered  the  universe  would  soon  become  a  single  solid  of 
infinitesimal  dimension  and  infinite  density.  Centrifugal  force, 
on  the  other  hand,  always  and  insistently  tends  to  separate  and 
diffuse  mass ;  and  if  it  had  its  way  unhindered  the  universe  would 
soon  become  a  gas  of  infinite  volume  and  infinitesimal  density. 
But  neither  thing  happens.  So  far  as  we  can  see,  the  mean 
volume  and  the  mean  density  of  the  universe  both  remain  con- 
stant. The  simple  explanation  is  that  centripetal  and  centrifugal 
forces  are  always  paired,  in  counterbalance.  Sometimes  one  pre- 
vails, temporarily  and  locally,  and  sometimes  the  other.  But 
neither  is  ever  absent.  Neither  is  ever  completely  free  to  act. 
The  existence  of  either  one  alone  is  unknown  in  nature,  is  in- 


MECHANICAL  WERGJF*       *J  13 

%£*UF2*2^< 

conceivable  to  the  naturally  taught  mmar*7m^=sftould  never  be 
mentioned  to  the  student  as  a  natural  possibility. 

Oi  these  two  fundamental  mechanical  tendencies  let  us  con- 
sider first  the  centripetal  one. 

It  may  be  regarded  as  the  prime  fact  of  nature  that,  except 
when  an  excess  of  motion  disguises  the  fact,  all  things  are  obvi- 
ously bonded  together  by  a  mutually  attractive  force.  The  "law" 
— or,  better,  the  fact — of  gravitation  is  simply  the  sublime  condi- 
tion that  each  two  bodies  in  the  universe,  of  whatever  sort,  at 
whatever  places  and  at  all  times,  are  drawn  toward  each  other 
by  a  mutual  tie  of  affection — an  affection  so  true  and  unvarying 
that  to  liken  it  to  mere  human  affection,  which  is  always  partial 
and  fickle,  were  belittling  it  indeed.  This  tie  can  never  be 
broken,  although  it  can  be  stretched  indefinitely. 

The  old  saw  has  it  that  it  takes  two  to  make  a  quarrel.  Well, 
it  takes  two  to  make  a  bond  of  affection,  just  as  well;  and  it 
takes  two  mass-portions  to  exert  a  gravitational  attraction.  As 
Newton  defined  the  law,  more  than  two  centuries  ago,  the  force 
between  each  pair  of  masses  varies,  first,  directly  as  the  product 
of  their  masses,  and,  secondly,  inversely  as  the  square  of  their 
distance.  Stated  mathematically  this  becomes 

Force =cM1M2-^-,  (1) 

wherein  the  M's  represent  the  masses  concerned,  S  their  dis- 
tance of  separation  and  c  a  constant.  When  the  masses  are 
measured  in  units,  each  of  which  weighs  32.16  pounds,  on  the 
earth's  surface,  and  when  S  is  stated  in  feet,  the  value  of  c,  in 
order  that  the  force  shall  also  read  in  pounds,  becomes  about 
0.000,000,0343,  or  one  divided  by  about  twenty-nine  or  thirty 
millions.* 

If,  in  this  formula,  M±  should  be  stated  as  the  mass  of  our 
earth  and  S  as  its  radius,  c  Mj-f-S2  would  reduce  to  32.16 
pounds;  or  the  gravitational  force  exerted  by  any  unit  mass  at 
the  earth's  surface  would  be  32.16  pounds.  If  M±  and  M2  were 
a  pair  of  steel  plates,  one  inch  thick  and  about  twenty-two  feet 
square,  hung  up  vertically  in  chains  in  contact,  face  to  face,  it 
would  require  a  force  of  about  one  pound  to  overcome  their 
mutual  gravitational  attraction  and  pull  them  apart. 

*These  figures  follow  Professor  Poynting,  as  quoted  by  Professor 
J.  J.  Thomson  in  Engineering  (London),  March  19,  1909,  page  397. 


14  ENERGY 

This  may  seem  like  a  very  small  force.  But  it  is  only  because 
either  one  of  the  mated  plates,  pulling  upon  its  brother,  is  a  very 
small  thing  in  comparison  with  the  earth  which  is  pulling  upon 
them  both.  It  is  also  to  be  remembered  that  the  force  increases, 
as  the  masses  grow  larger,  by  their  product,  and,  as  the  distance 
grows  smaller,  by  the  reciprocal  of  its  square.  Thus  it  becomes 
plain  how  this  force  may  become  sufficient,  upon  occasion,  to 
hold  in  position  the  gigantic  heavenly  bodies,  on  the  one  hand,  or 
so  to  bind  together  the  minute  particles  of  steel,  on  the  other, 
that  a  stress  of  100,000  pounds  per  square  inch  cannot  pull  them 
apart. 

But  force  without  space  does  not  constitute  energy.  It  is 
only  as  the  separation  alters  that  energy  appears.  Multiplying 
the  force,  therefore,  by  a  small  change  in  distance  (dS,  in  the 
calculus)  and  integrating,  there  results 

Potential  Energy  =  Ep  =  c  M,Ma  (4-  -  4-)  =  c  M1M2  (-^^-)   (2) 

o         b  bb 


wherein  S0  is  any  original  distance  of  separation  and  S  any 
other.  As  in  Equation  i,  if  S  be  measured  in  feet  and  M  in 
units  weighing  g  pounds  the  result  will  be  a  value  for  Ep  in 
foot-pounds. 

If,  further,  the  quantity  S  —  Sc  be  given  the  symbol  h,  Equation 
constant,  and  if  S—  S0  be  very  small  in  proportion  to  S,  then 
c  Mj/SSo  may  be  taken  as  a  constant  and  given  the  symbol  g. 
If  further,  the  quantity  S  —  S0  be  given  the  symbol  h,  Equation 
2  becomes 

Ep  =  gM2(S-S0)  =  Wh  (3) 

which  is  the  formula  for  potential  energy  more  familiar  to  engi- 
neers than  the  correct  one,  Equation  2.  But  Equation  3  now 
appears  in  its  proper  light,  viz:  as  a  mere  approximation  which 
suffices  in  accuracy  under  certain  assumed  special  conditions  only. 
The  same  general  aspect  of  the  situation  applies  also  to  the 
question  of  kinetic  energy.  If  a  moving  body  be  accelerated, 
either  positively  or  negatively,  force  is  manifested  and  energy 
developed.  But  to  this  phenomenon  the  participation  of  at  least 
two  bodies  is  essential.  No  moving  body  can  be  accelerated 
except  by  reaction  against  a  second  body.  We  have  Newton's 
own  word  for  that.  It  is  quite  erroneous  to  say  that  any  moving 
body  possesses  energy  by  reason  of,  and  to  the  extent  of,  its 


MECHANICAL  ENERGY  15 

visible  motion.  Relative  motion  between  two  bodies  can  be  per- 
ceived visibly  quite  independently  of  their  relative  mass.  We 
can  measure  the  motion  of  distant  suns  from  a  base  of  observa- 
tion, to  wit,  the  earth,  so  small  that  the  sun  in  question  would 
never  perceive  it  if  it  hit  us  and  wiped  us  out  of  existence.  But 
the  measure  of  the  kinetic  energy  between  two  moving  bodies 
depends  altogether  upon  the  mass  of  the  point  of  reference.  For 
a  moving  body  possesses  energy  only  to  the  extent  that  it  can  be 
stopped.  And  that  extent  is  settled,  not  by  the  original  body 
alone,  but  by  the  mass-relationship  existing  between  motor  and 
arrestor. 

If  any  mechanic  doubts  this  statement,  let  him  try  to  forge  a 
bolt  upon  an  anvil  made  of  wet  clay,  which  cannot  bring  its 
entire  mass  promptly  to  the  job  of  arresting  the  hammer.  Or 
let  him  try  it  even  with  a  steel  anvil,  but  one  having  a  mass  only 
equal  to  that  of  his  hammer.  He  will  find  that  his  hammer,  and 
the  muscles  which  yesterday  drove  it  with  effective  energy,  to-day 
are  powerless.  The  muscles  are  vigorous  and  the  hammer-head 
lively;  but  action  can  be  no  greater  than  reaction.  As  the  anvil 
reacts,  so  only  may  the  hammer  act.  Motion  may  be  there,  but 
not  necessarily  kinetic  energy. 

These  ideas  may  be  reduced  to  mathematical  expression  by 
starting  with  the  empirical  equation  of  motion, 

Force  =  Mass  x  Acceleration.  (4) 

From  this  there  results,  by  algebra  which  need  not  be  reproduced 
here,  the  following  fundamental  expression  for  the  kinetic 
energy  of  any  mass-pair  : 

E,  —  —      MiM2    ,V2    v  2.  ... 

^k~  2 


wherein  the  M's  are  the  two  mass-portions  (measured  in  units 
weighing  32.16  pounds  at  the  earth's  surface),  the  V's  are  the 
original  and  final  velocities,  in  feet  per  second,  respectively,  and 
Ek  is  the  energy  in  foot-pounds.* 


*  The  reader  may  be  assisted  to  connect  these  fundamental  formulae 
for  energy  with  his  more  familiar  engineering  concepts  by  the  following: 

Let  Fig.  A  represent  two  bodies,  Mt  and  M3,  having  a  common  center 
of  mass  at  C.  Let  the  bodies  possess  a  motion  toward  or  away  from  C 


16  ENERGY 

If  M!  be  very  large  in  comparison  with  M2  the  fraction  re- 
duces to  approximately  the  value  M2.  If,  further,  V0  =  o,  the 
expression  becomes 

Ek— i-M  V2  (6) 

which  is  the  special  approximation  which  the  engineering  pro- 
fession generally  regards  as  the  true  fundamental  equation  for 
kinetic  energy. 

So  long  as  the  mind  confines  itself  to  engineering  problems 
this  special  approximation  serves  as  well  as  the  exact  expression. 
But  the  text-books  certainly  ought  never  to  leave  it  to  the  maga- 
zine press  to  inform  the  student  that  it  is  a  special  approximation, 
instead  of  the  true  article.  And  when  this  special  approximation 
is  carried  into  the  problems  of  molecular  energy,  where  it  is 
applied  as  the  sole  available  concept  of  kinetic  energy,  this  bit 

of  v,  and  va  respectively.  Then,  from  the  law  of  conservation  of  mo- 
mentums, 

M,vt  =  M2v3  (a) 

and  Vl  +  v3  =  V  (b) 

wherein  V  is  the  relative  velocity  between  the  two  bodies. 

If  this  relative  motion  be  opposed  by  a  force  of  magnitude 
Mtai  (=  M2a2),  acting  upon  each  mass  and  reacting  upon  the  other,  it 
will  be  destroyed  at  a  rate  of  negative  "acceleration"  equal  to 

A  =  ai  +  aa  (c) 

wherein  at  is  the  change  in  velocity  per  second  measured  between  MI  and 
C,  aa  is  that  measured  between  M3  and  C,  and  A  is  that  measured  between 
Mi  and  M3. 

When  the  motion  has  been  entirely  overcome  the  opposing  force  will 
have  been  overcome  by  the  body  Mi  through  the  distance  $  Vi,  and  by 

If, 


o- 


FIG.    A. 

the  body  M3  through  the  distance  \  v2,  or  over  a  total  distance  of  $  V. 
The  work  performed  will  have  been  equal  to  the  force  times  the  distance 
in  the  case  of  each  body,  or,  together, 

Work  =  M!V,  (J  vO  +Mava  Q  v3)  (d) 

From  Equations  a  and  b,  VM2 

V'  ~=    Mt  +  M3 

VM, 
and  v'  =  -  f 


The  substitution  of  Equations  e  and  f  in  Equation  d  gives 

2 


for  the  special  case  where  the  relative  motion  is  completely  destroyed. 


MECHANICAL  ENERGY  17 

of  careless  neglect  becomes  an  egregious,  fundamental  error, 
which  has  misled  many  an  able  mind  which  lacked  the  time 
needed  to  dig  out  the  straight  of  the  matter. 


The  significance  of  the  above,  aside  from  some  most  im- 
portant conclusions  which  will  be  drawn  from  it  in  later  pages, 
amounts  to  this : 

1.  Energy  exists  only  between  bodies,  and  never  in  them. 
No  single  body,  by  any  quality  assignable  to  it  as  a  unit,  can 
ever  possess  energy. 

2.  The  energy  frequently  spoken  of  as  internal  to  a  "single" 
body  implies  that  the  body  in  question  is  not  a  single  unit,  but  a 
swarm  of  tiny  particles,  each  separate  from  its  neighbors,  yet 
too  small  to  be  seen  separately,  between  which  exists  the  energy 
said  to  be  "internal"  to  the  body. 

3.  The  relationships  between  mass-portions  which  constitute 
energy  may  be  either  of  two  sorts,  viz :    j/>ar^-relationships  and 
fwofr'ow-relationships,  called  potential  and  kinetic,  respectively. 

4.  Neither  relative  space  nor  relative  motion  themselves  con- 
stitute energy,  but  only  change  in  either  space  or  motion.    There- 
fore, every  true  expression  for  energy  must  contain  the  differ- 
ence between  a  greater  and  a  lesser  value  for  space  or  motion,  as 
the  case  may  be.    And  it  is  a  general  fact  of  nature  that  nowhere 
is  the  smaller  measure  of  either  space  or  motion  ever  known  to 
become  zero. 

5.  Every  true  expression  for  energy  must  contain,  for  its 
mass-factor,  the  arithmetical  product  of  the  two  quantities  of 
mass  concerned. 

6.  The  elementary  or  unit  mass  factor  in  energy  is  not  a 
unit  of  mass,  as  is  now  universally  taught,  but  the  unit  mass-PAiR; 
that  is  to  say,  a  pair  of  mass-portions,  each  member  of  which  is 
one  unit  of  mass. 

7.  The  ability  of  different  mass  systems  to  embody  energy 
is  proportional,  not  to  the  total  mass  of  each,  but  to  the  square 
of  that  aggregate  mass.    This  is  necessarily  true  of  space-energy, 
but  may  or  may  not  be  true  of  kinetic  energy,  according  to  the 
nature  of  the  motions  contemplated. 


CHAPTER  II. 

FREE  AND  VIBRATORY  ENERGIES. 

In  the  preceding  article  energy  was  defined  as  a  change  in 
either  the  space-relationship  or  the  motion-relationship  between 
two  or  more  bodies.  It  is  necessary  now  to  see  what  sorts  of 
space  and  motion  relationships  are  possible  in  nature. 

Let  Fig.  i  represent  an  ordinary  pendulum  attached  to  two 
A-frame  supports  on  the  earth's  surface.  Here  exists  an  energy- 
system  in  which  both  space  and  motion  relationships  occur. 
Moreover,  assuming  the  pendulum  to  be  in  motion,  there  is  a 
constant  change  of  both.  Finally,  there  are  present  the  two 
separate  mass-portions  which  were  stated  in  the  preceding  paper 
to  be  essential,  between  which  the  energy  is  embodied.  One 
mass-portion  is  the  pendulum-bob  Mx  and  the  other  is  the 
earth  M2. 

If  Mt  be  held  stationary  at  A  the  system  exhibits  space- 
energy  only.  If  it  be  released  at  A,  however,  the  pendulum 
swings  toward  C,  speeding  up  as  it  goes.  But  at  C  it  has  reached 
its  maximum  velocity,  and  beyond  that  point  it  slows  down  until 
B  is  reached,  where  it  stops,  reverses  and  repeats  the  process  in 
reversed  order.  In  this  simple  and  familiar  phenomenon  is  ex- 
hibited one  of  the  broadest  and  most  impressive  principles  of 
natural  action,  viz: 

THE  CONSERVATION   OF   ENERGY. 

For  if  the  space-energy  lost  between  A  and  C  be  carefully  meas- 
ured, and  also  the  motion-energy  gained,  the  two  will  be  found 
to  be  equal.    Expressed  mathematically, 
MM.  2  _     .  ,  ,  , 

Va)        MlMaC  } 


2     Mi+M  a  la     Sc         S 

wherein  the  a-subscripts  refer  to  conditions  at  either  A  or  B,  and 
the  c-subscripts  to  those  at  C.  This  is  the  fundamental  equation 
for  energy-transformation. 

Fig.  I,  however,  does  not  represent  an  ideal  or  perfect 
system,  because  the  connection  between  the  two  masses  is  estab- 
lished by  means  of  a  nominally  flexible  cord  or  hinge,  which  in 

18 


FREE  AND  VIBRATORY  ENERGIES  19 

actuality  always  involves  friction.  Nor  may  it  be  inferred  from 
this  that  the  words  ideal  and  perfect  refer,  as  they  usually  do 
when  used  in  connection  with  heat-engines,  or  gases,  to  some 
imaginary  and  impossible  conditions,  never  found  in  nature. 
For  in  mechanics,  while  there  is  no  known  instance  of  force 


FIG.   I. 

being  transmitted  from  body  to  body  by  contact  without  some 
loss  of  energy  in  friction  and  impact,  yet  it  is  common  for  such 
to  occur  by  action  at  a  distance.  Indeed,  there  exist  no  two 
bodies  in  the  universe,  according  to  Newton,  between  which  force 
is  not  thus  perfectly  transmitted,  by  gravitational  attraction,  and 
between  which  motion  cannot  occur  under  forceful  control,  yet 
without  friction.  This  fact  is  the  very  foundation  of  our  entire 
science  of  mechanics.* 

*This  statement  takes  entire  cognisance  of  the  resistance  to  the  motion 
of  celestial  bodies  through  interstellar  space  which  has  been  revealed  by 
modern  astronomy.  The  fact  of  this  resistance  upholds  all  the  positions 
taken  here  and  in  later  pages,  as  to  our  fundamental  concepts  of  true 
mechanics,  celestial  or  molecular.  The  laws  of  Newton  and  Kepler,  and 
the  fundamental  mechanical  principles  which  have  been  deduced  therefrom, 
are  all  based  upon  the  assumption  that  the  celestial  bodies  move  through 
matterless  space.  They  hold  true  only  in  that  case.  Indeed,  our  only 
idea  of  a  "body"  is  a  portion  of  matter  separated  from  other  portions  of 
matter  by  true  space,  in  which  exists  no  matter.  Later  in  these  pages  it  is 


20  ENERGY 

The  use  of  this  gravitational  action  at  a  distance  to  link  two 
bodies  together  into  an  energetic  system  similar  to  Fig.  I  is  shown 
in  Fig.  2,  as  a  diagram  of  an  energy-transforming  system  which 
would  remain  in  operation  indefinitely,  with  its  energy  perfectly 
conserved ;  because  it  .relies  solely  upon  "action  at  a  distance,"  or 
force  transmitted  without  friction.  In  Fig.  2  one  of  the  masses, 
M2  (the  earth,  if  you  please),  is  shown  pierced  centrally  with  a 
great  well-hole — much  as  Captain  Symmes  imagined  the  earth 

M, 

A 


FIG.    2. 

to  be  from  pole  to  pole,  in  his  long-ago  famous  "Symmes  Hole" 
theory.  Above  the  earth's  surface  and  in  line  with  this  hole,  as 
at  A,  is  suspended  the  body  Mx.  Upon  its  release  it  will  vibrate, 
quite  as  did  the  M!  of  Fig.  I,  on  either  side  of  the  point  of 
closest  approach  to  M2  at  C. 

To  this  system  Equation  7  applies  perfectly.  In  this  case,  too, 
occurs  good  illustration  of  how  the  lesser  distance  of  separation 
Sc  never  becomes  zero,  although  at  the  point  C  the  geometric 
centers  of  the  two  spheres  happen  to  coincide.  But  in  energetics 


stated,  as  a  broad  natural  empiricism,  that  there  is  no  such  a  thing  as  true 
"space"  devoid  of  matter — that  some  degree  of  mass,  pressure,  temperature, 
etc.,  exists  everywhere.  This  is  upheld  by  the  fact  of  resistance  to  celestial 
motion  showing  that  the  celestial  bodies  move  through,  not  space,  but 
attenuated  matter,  probably  interspersed  with  small  solid  bodies.  It  will 
also  be  pointed  out  that  there  exists  in  nature  no  such  a  thing  as  true 
matter;  that  is,  matter, devoid  of  space. 

All  this  does  not  affect  the  fact  that  we  are  now  two  centuries  gone  on 
a  course  starting  from  the  concept  of  true  matter  and  true  space,_  as  con- 
trasted absolutes.  Every  boy's  experience,  every  student's  training  in 
elementary  mechanics  and  every  engineer's  instinctive  judgment  are  based 
upon  this  same  idea,  which  is  also  the  foundation  of  the  Newtonian 
mechanics.  The  writer  is  merelv  insisting  that,  since  the  doctrine  of  all 
mechanics  as  founded  on  the  Newtonian  concepts,  and  of  heat  as  being 
"a  mode  of  motion,"  is  promulgated  universally  in  all  engineering  schools, 
its  consequences  must  be  faced  with  consistency. 


FREE  AND  VIBRATORY  ENERGIES  21 

it  is  only  mass  which  counts,  not  geometry.  Although  the  geo- 
metric centers  may  coincide  at  C,  the  two  masses  do  not.  After 
M!  enters  the  geometric  boundaries  of  M2  their  real  separation 
decreases  only  gradually,  until  at  C  it  is  a  minimum,  but  not  zero. 

It  happens,  however,  that  the  case  illustrated  in  Fig.  2.  never 
occurs  in  nature.  It  is  not  that  the  major  mass-portions  of  the 
universe  neglect  to  possess  convenient  openings  for  the  passage 
of  the  minor  mass-portions  through  them.  The  trouble  is  that  in 
nature  the  chances  are  infinitely  against  any  pair's  ever  possessing 
a  motion,  when  in  the  condition  A,  which  is  directly  alined 
with  their  mutual  centers.  The  thing  is  conceivable  geo- 
metrically, but  it  is  even  less  than  probable.  When  the  natural 
causes  of  different  forms  of  motion  are  investigated,  it  will 
appear  that  such  motion  is  impossible.  Motion  developed  nat- 
urally, rather  than  in  the  imagination,  will  be  directed  at  some 
appreciable  angle  with  the  mutual  axis,  as  at  A  in  Fig.  4. 

In  any  natural  case  the  mutual  motion  will  be  likely  to  assume 
the  form  portrayed  in  Fig.  3 — to  consider  the  simplest  case  first. 
Here  the  bodies  are  shown  as  revolving  about  each  other,  and 
also  about  their  common  center  of  mass  at  C.  Each  body  de- 
scribes an  elliptical  orbit  about  its  mate,  and  also  about  the 
point  C.  For  a  moment's  consideration  will  show  that,  whatever 
form  of  orbit  one  body  follows  around  the  other,  the  other  must 
likewise  describe  about  the  one;  and  if  the  mass-center  C  be 
regarded  as  the  fixed  center,  each  body  will  describe  about  it 
similar  orbits,  having  radii  inversely  proportional  to  the  mass  in 
question. 

The  situation  is  much  as  if  a  boy  fastened  a  large  cannon- 
ball  on  one  end  of  a  stick  and  a  smaller  one  on  the  other.  He 
can  then  twirl  the  stick  by  holding  either  end  in  his  hand,  re- 
garding that  as  fixed  while  the  other  end  moves,  or  he  can  hold 
the  middle  portion  of  the  stick  in  his  hand,  at  the  center  of 
gravity,  and  twirl  both  ends  at  once  about  that.  But  in  the  case 
of  the  boy  and  the  stick,  his  hand  is  attached  to  the  earth,  and 
may  be  regarded  as  fixed;  whereas  in  the  case  of  the  two  free 
mass-portions,  if  we  are  to  study  their  own  interaction,  inde- 
pendently of  all  other  masses,  there  is  no  fixed  base  to  refer 
anything  to.  Should  the  boy  throw  the  stick  into  the  air,  how- 
ever, for  a  brief  period  it  would  act  freely  as  an  independent, 
free  mass-system,  and  during  that  time  the  two  cannon-balls 


22  ENERGY 

would  revolve,  each  about  their  common  center  of  mass.  For 
that  reason  it  is  proper  to  consider  the  energetic  action  of  the 
members  of  a  pair  only  in  reference  to  their  common  center  of 
mass. 

In  any  such  an  elliptic  orbit  the  condition  of  greatest  separa- 
tion, as  at  ab,  is  called  the  apastron  of  the  orbit,  while  that  of 
least  separation,  as  at  AB,  is  called  the  periaston.  At  apastron 
the  energy  existent  between  the  two  is  chiefly  space-energy,  but 
there  is  also  a  little  motion.  At  periastron  the  energy  is  chiefly 


FIG.  3. 

motion-energy,  but  there  is  also  a  little  space  present.  The 
equation  which  connects  mathematically  these  maxima  and 
minima  of  space  and  motion  is  Equation  7 ;  and  in  it,  in  all  truly 
natural  cases,  neither  V0  nor  S0  may  ever  be  regarded  as  be- 
coming equal  to  zero — as  will  be  made  plain  as  the  argument 
proceeds. 

For  Fig.  3 — or,  rather,  its  more  general  form,  Fig.  4 — repre- 
sents the  only  true  element  of  mechanical  action.  For  any  such 
an  element,  in  order  to  be  an  element,  must  be  absolutelv  "free." 
That  is  to  say,  it  must  be  considered  independently  of  all  other 


FREE  AND  VIBRATORY  ENERGIES  23 

masses.  Yet  it  must  be  capable  of  containing  energy.  There- 
fore, it  must  be,  not  an  ultimate  or  indivisible  unit  of  mass,  but  a 
mass-pair,  an  elementary  mass-pair. 

All  action  between  solid  bodies  in  contact,  on  the  other  hand, 
such  as  is  familiar  to  all  engineers  in  their  machines,  is  not 
purely  mechanical.  It  is  always  partly  thermodynamic,  in  so  far 
as  heat  is  being  constantly  developed  by  friction,  and  partly  a 
special  case  of  pure  mechanics,  in  that  the  body  is  "constrained" 
rather  than  free;  that  is,  it  is  handling  energies  which  are 
transient  through  it  from  without,  which  are  independent  of  its 
own  mass,  and  which  are  ultra-complex  in  their  nature. 

Fig.  4,  on  the  other  hand,  is  entirely  general.  It  displays 
every  possible  form  of  pure  and  elementary  mechanical  action 
between  two  bodies,  supplied  with  any  original  store  of  relative 
space  and  relative  motion  whatever,  as  at  A;  and  it  introduces 
no  foreign  element  of  dependence  upon  any  other  mass-system 
or  form  of  energy  whatever.  Nor  does  it  introduce  any  un- 
natural assumptions.  Without  stopping  now  for  the  proofs,  it 
may  be  said  that  any  such  a  case  must  resolve  itself  into  the 
mutual  revolution  of  the  bodies  about  each  other  in  an  orbit 
which  follows  some  of  the  plane  conic  sections — either  the 
hyperbola,  the  parabola,  the  ellipse,  the  circle  or  the  straight  line. 

Further,  it  can  be  shown  that  if  the  original  energetic  condi- 
tion of  the  pair  at  A  M2  be  known,  by  knowledge  of  the  dis- 
tance d  between  the  two,  the  velocity  v  of  their  relative  motion, 
the  angle  <£  existent  between  d  and  v,  and  the  two  masses  Mt  and 
M2,  then  the  nature  of  the  orbit  is  known,  and  also  its  dimen- 
sions. Both  are  best  expressed  in  terms  of  the  distance  D  be- 
tween the  two  when  separated  by  a  radius  normal  to  the  major 
axis  XX'  of  the  orbit,  the  velocity  U  at  that  point,  and  the  angle 
a  between  D  and  U.  The  mathematical  relationships  between 
all  these  quantities  will  be  discussed  later. 

Of  all  these  apparently  varied  forms  of  motion  only  two,  the 
hyperbola  and  ellipse,  are  probable  forms.  For  between  any  two 
masses,  at  any  given  initial  distance,  there  may  be  an  infinite 
number  of  directions  and  magnitudes  of  velocity  which  would 
result  in  hyperbolic  motion,  and  another  infinite  number  which 
would  result  in  elliptic  motion.  But  there  is  only  a  single  direc- 
tion of  motion  which  would  result  in  a  straisrht-line  orbit,  such 
as  that  of  Fig.  2 :  and  there  is  only  one  other  direction  of  motion, 


24 


ENERGY 


with  a  particular  velocity  into  the  bargain,  which  would  result 
in  circular  motion.  A  parabolic  orbit  could  likewise  result  from 
only  a  single  direction  of  motion,  coupled  with  its  proper  mag- 
nitude. Therefore  the  mathematical  chances  are  infinitely  in 
favor  of  all  motion  in  the  universe  being  either  elliptic  or  hyper- 
bolic in  form. 

The  possibilities,  however,  are  not  even  so  complex  as  even 
this  statement  might  indicate.  Speaking  mathematically,  there  is 
for  all  of  these  forms  of  orbit  but  a  single,  basic  equation.  This 
equation  must  involve  some  factor  expressive  of  radial  distance 
between  the  two,  and  also  some  factor  expressive  of  either  the 


FIG.   4. 

two  quantities  of  mass  or  their  periodic  velocity  of  revolution. 
These  will  define  the  general  magnitude  of  the  system.  But 
besides  these  factors  there  must  also  be  present,  as  an  indicator 
of  the  form  of  orbit,  another  which  is  called  the  eccentricity. 
And  the  eccentricity  of  orbit,  it  will  be  found,  is  a  factor  of 


FREE  AND  VIBRATORY  ENERGIES  25 

fundamental  importance  in  the  determination  of  whether  energy- 
transformation  is  to  occur  or  not. 

Without  attempting  to  enter  into  any  discussion  of  the  mathe- 
matical definition  of  eccentricity,  which  is  usually  given  the 
symbol  e,  the  following  characteristic  facts  should  be  noted  care- 
fully, as  indicating  sufficiently  for  our  purposes  its  general 
nature : 

1.  If  the  eccentricity  be  zero  the  orbit  is  a  circle; 

2.  If  the  eccentricity  be  greater  than  zero,  but  less  than 
unity,  the  orbit  is  an  ellipse ; 

3.  If  the  eccentricity  be  unity  the  orbit  is  a  parabola ; 

4.  If  the  eccentricity  be  greater  than  unity,  but  finite,  the 
orbit  is  an  hyperbola; 

5.  If  the  eccentricity  be  infinite  the  orbit  is  a  straight  line. 

Of  all  possible  cases,  therefore,  the  circle  constitutes  one 
extreme  and  the  straight  line  the  other.  Mathematically  speak- 
ing, they  constitute  mathematical  limits,  with  the  chances  infi- 
nitely against  their  ever  being  attained  in  nature.  But,  speaking 
naturally,  this  means  that  they  never  occur.  Both  zeros  and  in- 
finities are  unimaginable,  as  natural  phenomena.  They  have 
never  been  observed,  and,  so  far  as  the  human  mind  may 
predict,  they  never  will  be  observed.  It  is  one  of  the  heaviest 
indictments  to  be  brought  against  our  present  methods  of  teach- 
ing natural  science  that  its  text-books  are  so  filled  with  reckless 
use  of  zeros  and  infinities.  It  is  not  that  zeros  and  infinities 
should  never  be  employed,  but  that  they  should  always  be  speci- 
fied, before  using,  as  natural  impossibilities — as  is  always  done, 
for  instance,  in  specifying  the  exclusion  of  friction,  thermal  con- 
duction, etc. 

As  for  the  parabolic  orbit,  that  plainly  stands  as  the  dividing 
case  between  the  ellipses  below  and  the  hyperbolas  above.  It 
itself,  like  the  circle  and  the  straight  line,  is  infinitely  improbable 
of  occurrence,  in  any  permanence  of  form.  But  whereas  the 
chances  are  infinite  against  truly  circular  or  straight-line  motion 
ever  being  attained,  even  instantaneously,  the  condition  of  para- 
bolic motion  must  be  at  least  crossed,  and  therefore  existent  in- 
stantaneously, in  transition  from  elliptic  to  hyperbolic  motion. 
For  the  parabolic  condition  is  like  a  dividing  line  between  two 
areas:  having  no  dimension,  substance  or  reality  itself,  it  must 


26  ENERGY 

nevertheless  be  encountered  and  crossed  by  substantial  realities, 
in  their  passage  from  one  territory  to  the  other. 

But,  it  will  naturally  be  asked,  what  have  all  these  astronom- 
ical illustrations  of  "free"  bodies  to  do  with  engineering  me- 
chanics, when  no  engineer  or  mechanic  ever  saw  anything  in  his 
work  which  acts  in  that  way  ?  The  answer  is  that,  in  spite  of  the 
strangeness  of  the  idea,  this  is  the  only  way  in  which  any  portion 
of  matter  ever  really  does  act,  when  it  acts  purely  mechanically. 
In  some  departments  of  mechanics,  chiefly  in  gunnery,  the  pro- 
jectiles which  form  the  chief  subject  of  study  do  come  very 
nearly  to  acting  in  this  way.  The  only  assumption  which  has  to 
be  made,  in  reducing  their  ordinary  action  to  a  true,  or  "free,"  or 
natural  basis,  is  that  the  friction  of  the  atmosphere  be  absent. 
Then  their  paths  are  commonly  said  to  be  parabolic.  But  this  is 
a  statement  which  is  only  approximately  true.  It  is  made  be- 
cause it  is  simple  and  easy  to  assume  that  the  lines  of  gravita- 
tional force  which  radiate  from  the  earth's  center  are,  for  small 
portions  of  its  surface,  virtually  parallel;  and  also  because  the 
result  is  not  far  enough  wrong  to  upset  calculations. 

But  for  the  special  purposes  of  our  argument  it  is  far  better 
to  keep  in  mind  the  exact  truth — which  is  simple  enough  so  long 
as  we  do  not  try  to  make  any  detailed  calculations — namely,  that 
all  cannon-balls,  base-balls,  foundry-drops,  etc.,  are  in  reality 
following  elongated  elliptical  orbits  passing  closely  around  the 
center  of  the  earth.  But  of  this  orbit  the  only  portion  which 
we  can  see  is  the  apastron  tip.  The  remainder  of  the  orbit 
never  is  traversed,  because  collision  with  the  solid  surface  of 
the  earth  breaks  up  the  phenomenon,  in  a  dissipation  of  energy 
in  thermodynamic,  rather  than  mechanical,  action.* 

But  even  to  get  such  a  slight  peep  as  the  above  at  pure  me- 
chanical action  on  the  surface  of  the  earth,  we  have  had  to 

*The  foundry-drop  is  ordinarily  regarded  as  falling  toward  the  earth  in 
a  straight  vertical  line.  Yet,  in  reality,  as  it  leaves  the  latch  it  possesses 
a  tangential  velocity,  parallel  with  the  earth's  surface,  of  over  fifteen  hun- 
dred feet  per  second,  due  to  the  earth's  rotation.  If  we  could  imagine  the 
mass  of  the  earth  as  suddenly  concentrated  at  its  center,  or  within  a  sphere 
about  ten  miles  in  diameter,  the  foundry-drop  would  be  free  to  describe  an 
elliptic  orbit  passing  around  the  center  of  the  earth  at  a  distance  of  about 
seven  miles,  and  exhibiting  at  periastron  a  tangential  velocitv  of  170  mile1? 
per  second.  Midway  toward  the  center  of  the  earth  this  ellipse  would 
snread  out  to  a  conjugate  diameter  of  over  one  hundred  mi^s.  Yet  this 
illustration  is  the  nearest  approach  to  rectilinear  motion  which  we  can 
produce  in  the  engineering  arts ! 


FREE  AND  VIBRATORY  ENERGIES  27 

make  the  never  true  assumption  that  the  friction  of  the  atmos- 
phere were  absent.  Yet  in  all  the  mechanical  actions  which  are 
even  more  familiar  to  the  engineer,  such  as  the  interaction  of 
solid  machine-parts,  see  what  even  more  wholesale  assumptions 
have  to  be  made,  in  order  to  weed  out  the  non-mechanical 
phenomena  of  friction  and  impact  and  get  a  glimpse  of  the  pure 
mechanics  behind  them !  It  is  scarcely  worth  while  to  try.  And 
yet  it  is  none  the  less  true  that  there  is  never  a  hammer-head, 
piston  or  shuttle  started  into  motion  that  it  does  not  try  to 
follow  an  elliptic  orbit  about  the  earth's  center;  from  which  it 
is  constrained  only  by  constant  supplies  of  energy  from  without, 
in  the  form  of  solid  forces  not  dependent  upon  the  masses  trans- 
mitting them,  which  forces  and  energies  are  called  "transient." 

For  taking  the  time  to  study  and  understand  these  free  or 
natural  tendencies  of  mass,  which  get  so  little  chance  to  display 
themselves  here  upon  the  earth's  surface,  there  are  two  reasons. 
One  of  these  reasons  is  the  fact  that  they  constitute  the  only 
true  mechanics,  from  which  all  engineering  happenings  are  but 
special  departures  and  for  which  only  approximate  statements 
can  be  made.  It  is  therefore  the  soul  of  true  education  to  teach 
these  exact  truths  first,  displaying  their  useful  applications  after- 
ward in  their  proper  aspect. 

The  other  reason  is  that  no  concept  of  heat  as  a  mechanical 
phenomenon  can  possibly  be  attained  without  them;  for  in  the 
mechanical  interaction  between  the  particles  of  a  body,  by  which 
we  must  now  attempt  to  explain  not  only  heat,  but  surely  also 
chemical  action  and  probably  also  electrical  and  magnetic  ener- 
gies, there  can  enter  no  friction  nor  impact.  The  action  must 
be  purely  mechanical.  There  can  be  no  dissipation  or  degrada- 
tion of  energy  into  heat,  because  it  is  heat  itself  which  is  being 
considered.  Nor,  as  stated  in  the  last  paper,  is  it  any  explanation 
of  the  puzzle  to  specify  that  the  particles  shall  be  perfectly 
elastic;  because  perfectly  elastic  matter  is  unknown  in  nature. 
It  is  only  space,  devoid  of  solid  contact,  which  is  perfectly 
elastic  in  nature;  and  the  first  task  in  the  comprehension  of 
heat,  therefore,  is  to  comprehend  thoroughly  this  "free"  depart- 
ment of  natural  action,  in  which  friction  and  impact  are  unknown 
and  unimaginable. 

The  task  of  the  theorist,  in  fact,  is  not  so  much  to  explain 
action  at  a  distance  in  terms  of  action  by  contact,  as  is  so  often 


28  ENERGY 

assumed,  but  fairly  the  reverse.  The  obscure  and  intricate  hap- 
penings, which,  in  our  ignorance  we  lump  together  under  the 
convenient  blanket-term  ''contact,"  involving  always  interchanges 
of  pressure  and  heat,  and  often  mechanical,  chemical  and  elec- 
trical energy,  none  of  which  we  understand,  can  be  explained 
simply  and  clearly  only  when  they  are  reduced  to  terms  of 
action  at  a  distance.  For  the  latter  demands  no  "explanation/'  or 
reduction  to  terms  of  something  else.  It  is  beautifully  simple, 
having  been  defined  in  an  elementary  algebraic  formula  by 
Newton  two  centuries  ago.  It  is  complicated  by  no  questions  of 
elastic  pressure,  thermal  or  electrical  conduction,  or  chemical 
interaction,  varying  interminably  with  each  new  case  of  "con- 
tact." Centripetal  gravitation,  like  its  mate,  centrifugal  force,  is 
one  of  the  basic  facts  of  the  universe — more  basic  even  than 
matter,  the  existence  of  which  we  infer  from  its  gravitational  and 
centrifugal  action — and  forms  no  proper  food  for  further  analysis 
until  it  shall  appear  to  us  in  much  more  intricate  guise  than  that 
which  we  inherit  from  Newton. 

Our  sole  duty  here  is  to  recognize  that  "contact"  is  merely  a 
convenient  name  for  impact  and  friction,  when  we  are  discussing 
mechanics,  for  thermal  conduction  when  we  are  discussing  heat, 
for  electrical  conduction  when  we  are  discussing  electricity,  etc. 
"Action  at  a  distance"  implies  merely  the  absence  of  any  of  these 
energetic  transformations. 


These,  then,  are  the  elementary  laws  of  motion,  when  viewed 
from  the  standpoint  of  energetics  rather  than  kinematics.  Since 
they  appear  to  be  quite  different  from  the  laws  of  motion  of 
Newton,  with  which  every  student  is  familiar,  while  apparently 
of  an  elementary  importance  equal  to  those  of  Newton,  it  is 
important  to  note  that  they  are  in  reality  the  laws  of  Newton, 
but  stated  in  combination  with  each  other  and  with  the  laws 
discovered  by  Kepler  some  sixty  years  before  Newton  enunciated 
his  law  of  gravitation  in  1680.  Newton's  laws  of  motion,  in 
their  familiar  form,  constitute  one  exceedingly  simple  form  in 
which  the  elements  of  mechanics  may  be  expressed.  The  trouble 
with  their  form  is  that,  in  order  to  get  each  statement  into  its 
simplest  possible  form,  a  set  of  premises  has  been  assumed 
which  is  peculiar  to  that  particular  statement  only,  and  which 


FREE  AND  VIBRATORY  ENERGIES  29 

not  only  never  occurs  in  nature,  but  which  is  directly  incon- 
sistent with  some  other  of  this  same  set  of  laws. 

For  instance,  one  of  these  laws  of  Newton  asserts  that  "a 
body  once  in  motion  will  continue  in  unchanging  straight-line 
motion  until  interfered  with  by  a  force."  But  Newton's  own 
law  of  gravitation  declares  that  no  body  in  the  universe  may 
ever  dissociate  itself  so  remotely  from  all  others  as  to  be  free 
from  interference  by  forces,  and  by  an  infinite  number  of  forces 
at  once.  Therefore,  unchanging  or  straight-line  motion  can 
never  occur  in  nature.  For  some  special  problems  of  mechanics 
it  has  been  useful  to  assume  that  it  could.  But  the  wholesale 
manner  in  which  this  special  and  temporary  assumption  has  been 
permitted  to  usurp  the  place  of  a  fundamentally  correct  and 
permanent  principle  of  nature  has  undermined  our  entire  under- 
standing of  the  problem  of  energetics. 

Again,  for  instance,  "a  body  subject  to  a  constant  force  will 
experience  constant  acceleration."  But  again,  Newton's  own 
law  of  gravitation  declares  that  force  can  remain  constant  only 
so  long  as  the  separation  between  the  bodies  remains  constant. 
But  this  is  impossible,  motion  being  assumed  at  all,  except  when 
the  motion  has  the  form  of  a  circular  orbit,  at  constant  distance ; 
in  which  case  the  motion  and  force  would  be  at  right-angles  with 
each  other,  and  mutually  independent.  In  all  other  forms  of 
motion  the  force  must  always  be  varied  by  the  motion  which  it 
itself  produces.  Therefore,  constancy  of  acceleration,  whether 
positive  or  negative,  is  unknown  in  nature. 

To  understand  properly  Newton's  laws  of  motion,  therefore, 
they  must  be  coupled  with  each  other  and  with  the  laws  of 
Kepler — upon  which  latter,  indeed,  they  were  founded.  Kepler's 
laws  are  three  in  number,  viz : 

1.  The  natural  path  of  free  motion  between  two  masses  is 
one  of  the  plane  conic  sections.* 

2.  The  area  swept  over  by  the  radius-vector  connecting  the 

*Kepler,  working  with  the  planetary  orbits  alone  as  his  material,  and 
quite  as  a  pioneer,  stated  this  law  originally  as  including  only  the  ellipse. 
It  is  later  learning  which  has  broadened  the  statement  to  include  all  of 
the  conic  sections.  A  very  simple  and  elegant  proof  of  this  law — depend- 
ing, however,  upon  Kepler's  Second  Law — was  published  by  Mr.  Immo  S. 
Allen,  of  the  London  Institution  CFinsbury  Circus,  London,  E.  C),  in  the 
Scientific  American  of  July  10th,  1909. 


30  ENERGY 

two  is  everywhere  equal  for  equal  periods  of  time;  or  the  "areal 
velocity"  is  constant. 

3.  The  square  of  the  orbital  period  (or  time)  of  revolution 
is  proportional  to  the  cube  of  the  major  axis  of  the  ellipse, 
divided  by  the  sum  of  the  two  masses. 

The  third  of  these  laws  again  illustrates  the  way  in  which 
fundamental  error  has  been  permitted  to  creep  into  the  standard 
college  text-book.  Of  all  of  the  ordinary  text-books  in  astronomy 
which  the  writer  has  happened  to  enter — not  to  find  fault,  but 
in  a  sincere  effort  to  straighten  out  this  tangled  question  of 
"what  is  energy" — only  one  mentioned  the  sum  of  the  masses  at 
all,  as  a  factor  in  Kepler's  third  law.  All  others  omitted  it, 
making  of  the  expression  for  the  third  law  a  special  approx- 
imation as  erroneous  as  is  the  formula  */2  M  V2  for  kinetic 
energy. 

Indeed,  until  this  single  case  was  discovered,  the  writer  was 
quite  puzzled  by  the  situation;  for,  according  to  all  he  knew  of 
the  elements  of  mechanics,  the  sum  of  the  masses  ought  to  appear 
in  the  statement  of  Kepler's  third  law.  And  yet  here  was 
standard  text-book  after  text-book  which  made  no  mention  of 
them!  Was  all  that  he  thought  he  knew  nonsense,  or  where 
else  was  the  trouble?  In  his  quandary  his  heart  went  out  to  all 
those  unfortunates  who  may  have  made  serious  attempt  to  under- 
stand the  general  principles  of  energetic  action  from  the  me- 
chanics taught  in  the  colleges  as  a  foundation. 

If,  now,  all  the  laws  of  Newton  be  kept  in  sight  at  once,  and 
those  of  Kepler  with  them,  the  elementary  principles  of  all 
mechanical  action  may  be  restated  as  follows.  In  their  literal 
form  they  apply  only  to  an  energetic  system  consisting  of  a 
single  pair  of  bodies.  In  the  sense  that  every  portion  of  the 
natural  universe  may  be — and,  according  to  any  philosophy 
founded  on  the  Newtonian  mechanics,  must  be — considered  as 
made  up  of  a  large  number  of  such  pairs,  with  its  distances, 
forces  and  motions  all  reducible  to  an  equal  number  of  com- 
ponents, one  for  each  pair,  they  are  universal  in  their  appli- 
cation. 

i.  Everywhere  is  space.  No  two  bodies  may  be  conceived 
as  coincident,  nor  any  one  body  as  occupying  zero  space.  The 
"occupation  of  space"  has  always  constituted  the  fundamental 
definition  of  "matter." 


FREE  AND  VIBRATORY  ENERGIES  31 

2.  All   space   is   relative,   measurable   only   between  bodies. 
Absolute  space  is  as  inconceivable  as  is  absolute  lack  of  space. 

3.  Everywhere  is  force.     Freedom  from  finite  force  is  un- 
known.    No  two  bodies  may  ever  become  so  widely  dissociated 
as  to  reduce  their  mutual  attraction  to  zero,  nor  so  closely  coin- 
cident as  to  raise  it  to  infinity. 

4.  All  force  is  relative.     It  exists  only  between  bodies,  and 
may  never  be  imagined  as  exerted  absolutely,  independently  of 
mass. 

5.  Everywhere  is  motion.     Absolute  rest,  or  fixity,  is  un- 
known and  inconceivable. 

6.  All  motion  is  relative.     Motion  is  measurable  and  con- 
ceivable only  between  the  members  of  a  related  pair.    No  single 
body,  independently  of  all  others,  may  possess  motion.    Absolute 
motion  is  as  inconceivable  as  is  absolute  rest. 

7.  Constancy  of  either  space,  force  or  motion  is  unknown  in 
nature.     Space  varies   force,   force  varies  motion,  and  motion 
varies  space,  all  the  time,  between  any  and  every  two  free  bodies. 

8.  Neither   straight-line   nor  circular   motion   is   known   in 
nature.     The  only  path  of  motion  which  is  natural,  rather  than 
imaginary,  hypothetical  and  superstitious,  is  either  the  ellipse  or 
the  hyperbola,  with  velocities  varying  as  stated  by  Kepler  and 
forces  varying  as  stated  by  Newton. 

9.  In   any  energetic   system  the  primary   fact   is   the   con- 
stancy, or  indestructibility  and  non-creatability,  of  its  mass.    The 
Principle   of    the   Conservation   of   Mass,    discovered    first   and 
needed  first,  should  certainly  receive  the  title  of  FIRST  LAW  OF 
ENERGETICS. 

10.  In  any  such  a  natural,  free  system  there  occurs  period- 
ically, at  each  revolution,  a  reversed  transformation  of  energy, 
from  space  to  motion  form  and  back,  under  the  Principle  of  the 
Conservation  of  Energy.     Discovered  only  in  1837 — after  much 
preliminary   investigation   and   partial   knowledge — and   not   yet 
fully  understood,  this  great  natural  principle  is  properly  to  be 
entitled  the  SECOND  LAW  OF  ENERGETICS,  not  the  "First,"  as  it 
is  now  called. 


CHAPTER  III. 

THE  MEAN  ENERGETIC  CONDITION  AND  THE  ENERGY-FUND. 

In  the  last  paper  attention  was  called  to  the  fact  that  all 
"free,"  or  natural  systems  of  mechanical  action  might  be  repre- 
sented by  some  one  of  the  conic  sections,  such  as  were  exhibited 
in  Figs.  3  and  4.  Of  these,  for  the  present  purposes  of  dis- 
cussion, the  elliptical  orbit,  as  shown  in  Fig.  3,  will  suffice  as  an 
illustration. 

In  elliptical  motion,  as  was  pointed  out  in  that  paper,  there 
occurs  a  periodic  energy-transformation  at  each  revolution  of  the 
bodies  about  each  other.  At  apastron  space  is  a  maximum  and 
motion  a  minimum;  at  periastron  the  reverse  is  true.  And  at 
every  point  between  these  extremes  the  conservation  of  energy 
is  maintained;  the  amount  of  either  form  of  energy  lost,  below 
the  maximum,  is  made  good  in  the  other  form. 

From  these  facts  it  is  readily  inferred  that  there  must  exist 
between  the  extremes  a  and  A  some  intermediate  point  where 
this  transformation  of  energy  from  space-form  to  motion-form, 
or  the  reverse,  is  just  half  accomplished.  Such  a  point  would 
constitute  a  true  energetic  mean  between  the  two  extremes.  At 
that  point  half  of  the  total  range  of  potential  energy  will  have 
been  converted  into  or  from  the  kinetic  form,  and  half  will  yet 
remain  to  be  converted. 

Let  us  indicate  the  distance  of  separation  between  the  two 
bodies  when  in  this  mean  energetic  condition  by  D,  and  their 
relative  velocity  by  U.  Then  there  must  result,  from  Equation  7 
for  the  conservation  of  energy, 

V>-IP)         (S) 


and  -^c  M,M2  (~-  -  -L)  =  c  M.M,  (-|-  -  -j)  (9) 

These  equations  readily  reduce  to 

u»~I-( 

32 


THE  MEAN  ENERGETIC  CONDITION  33 

cq 

and  .;  D  =  2^-|-       I  (11) 

In  the  case  of  elliptical  orbits  it  is  usual  to  represent  half  the 
major  axis  by  a,  half  the  minor  axis  by  b  and  the  eccentricity 
by  e.  In  that  case 

S  =  a(l+e)  (12) 

S0  =  a(l-e)  (13) 

D-2*rel-         ' 


b2 

But  —  is  one-half  of  the  latus  rectum  of  the  ellipse,  or  the 
a 

diameter  through  either  focus  at  right  angles  to  the  major  axis. 
That  is  to  say,  the  mean  energetic  distance  D  is  the  vector  or 
radius  joining  the  two  bodies  when  they  are  situated  directly 
at  right-angles  to  the  axis  joining  apastron  and  periastron. 

In  short  —  and  this  seems  to  be  most  important  —  the  energy- 
transformation  which  is  always  a  part  of  the  revolution  of  two 
bodies  about  their  common  center  of  mass,  is  a  function  of 
angular,  not  linear,  motion.  No  matter  how  eccentric  the  ellipse 
may  be,  it  is  always  true  that  in  each  quadrant  of  its  motion  the 
energy  is  just  one-half  transformed  —  from  extreme  space  or 
extreme  motion  to  the  mean  energetic  condition,  or  back. 

There  is  one  case  in  our  own  solar  system,  for  instance, 
where  one  quadrant  of  the  elliptic  orbit,  that  from  apastron  to 
the  mean  energetic  condition,  occupies  about  four  hundred  years 
and  covers  a  distance  measurable  in  hundreds  of  millions  of 
miles.  Yet  the  next  quadrant,  from  the  mean  energetic  condi- 
tion to  the  extreme  energetic  condition  nearest  the  sun,  trans- 
forming an  equal  quantity  of  energy,  occupies  only  a  little  over 
an  hour  and  covers  a  distance  measurable  in  thousandths  of  the 
other.  And  there  may  be,  of  course,  even  more  extreme  illus- 
trations of  eccentricity  of  orbit  than  this. 

In  Fig.  3  this  mean  energetic  position  is  shown  at  DD',  and 
in  Fig.  4  at  BM2.  In  Fig.  4  the  mean  energetic  velocity  U  is 
shown  as  maintaining  the  angle  a  with  the  vector,  or  latus 
rectum,  D.  It  will  be  convenient  to  note,  concerning  this  angle  a 
for  future  purposes,  that 


34  ENERGY 

e  =  — v/a2-b2  =cotana  (15) 

These  statements  and  arguments,  although  they  are  expressed 
most  simply  in  terms  of  the  ellipse,  could  be  established  as  apply- 
ing equally  to  any  of  the  conic  sections,  such  as  are  shown  in 
Fig.  4.  In  the  case  of  the  parabola  and  hyperbola,  which  have 
no  apastron  ends  and  no  definite  major  axes,  the  case  is  compli- 
cated (as  will  appear  later)  by  the  presence,  within  the  pair  of 
mass-portions,  of  two  funds  of  energy — the  one  which  has  just 
been  discussed  and  another.  For  the  present  it  will  be  quite 
sufficient  to  the  student  to  accept  on  faith  the  statements  that 
the  discussion  is  founded  upon  principles  which  are  general  in 
their  character,  applying  to  all  possible  cases. 

In  general,  then,  if  the  original  energetic  relationship  of  any 
pair  of  masses  whatever,  such  as  Mj  and  M2  of  Fig.  4,  be 
known — by  data  as  to  their  masses,  their  distance  d,  their  ve- 
locity v,  and  the  angle  <£  which  the  latter  makes  with  the  vector  d 
— all  the  conditions  of  the  orbit  and  the  energy-fund  embodied 
by  the  pair  are  known.  The  all-important  question  as  to  the 
amount  of  energy  thus  embodied,  which  is  a  very  difficult  one, 
will  be  resumed  at  a  later  point.  What  is  of  more  elementary 
concern  at  present  is  the  -form  of  the  energetic  action;  and  this 
depends  primarily,  it  is  obvious,  upon  the  eccentricity  of  orbit. 

It  is  worth  while  to  repeat,  in  a  briefer  form  and  with  in- 
clusion of  the  angle  a,  the  list  of  the  different  (mathematically) 
possible  types  of  orbit  which  was  given  in  the  preceding  paper. 

1.  If  e  =  0,  a  =  90°  and  the  orbit  is  a  circle. 

<  90° 

2.  If  e<  1,  a      ,r0  and  the  orbit  is  an  ellipse. 

3.  If  e  =  1 ,  a  =  45°  and  the  orbit  is  a  parabola . 

4.  If  e  >  1,  a<  45°  and  the  orbit  is  an  hyperbola, 

5.  If  e  =  3,  a  =  0°  and  the  orbit  is  a  straight  line. 

It  is  therefore  most  significant  that,  throughout  all  this  wide 
variation  in  values  of  e  and  x,  and  in  diversity  of  form  of 
orbit,  the  mean  energetic  condition  should  remain  consistently 
at  the  position  normal  to  the  orbital  axis.  Whatever  may  have 
been  the  original  distance  of  separation  from  which  the  two 
bodies  fell  together,  or  whatever  may  have  been  the  angle  swept 
over  by  the  vector  between  the  positions  A  and  B,  or  whatever 
may  be  the  speed  and  propinquity  at  which  the  bodies  pass  each 
other  at  periastron,  it  always  holds  true  that,  once  the  mean 


THE  MEAN  ENERGETIC  CONDITION  35 

energetic  condition  is  reached  and  one-half  the  energy-trans- 
formation accomplished,  at  B,  a  further  swing  of  just  one  quad- 
rant is  consumed  in  completing  the  other  half  of  the  energy- 
transformation,  to  periastron  at  P.  After  that  a  second  quadrant 
is  consumed  in  reversing  this  energy-transformation,  to  the  mean 
energetic  condition  again,  at  B' ;  after  which  the  cycle  ends  with 
a  reversal  of  the  angle  AM2B,  whatever  it  may  have  been — a 
quadrant  in  circle  or  ellipse,  or  less  than  a  quadrant  in  the 
parabola  or  hyperbola. 

Should  the  line  BB'  of  Fig.  4  be  considered  as  representing  a 
plane  normal  to  XX',  and  90°—  a  as  the  angle  of  incidence 
thereto,  the  angle  of  reflection  therefrom,  at  B',  will  always  be 
its  equal. 

The  Energy-fund:  Radial  and  Tangential  Energies.  If 
it  be  true  that  the  original  position  and  motion  of  the  pair 
at  any  original  condition,  such  as  A,  Fig.  4,  defines  all  the 
conditions  of  the  orbit,  it  should  be  possible  to  define  in  terms 
of  them  the  total  fund  of  energy  existent  in  the  pair.  Mathe- 
matically speaking,  this  is  possible — so  far  as  it  is  possible  to 
define  an  energy-fund  in  terms  of  any  premises  at  all.  But  the 
equations  which  connect  any  original  condition,  such  as  A,  with 
the  conditions  B,  P,  etc.,  are  so  cumbrous  in  form,  when  com- 
bined, that  it  is  not  practicable  thus  to  define  the  energy-fund  in 
terms  of  any  original  point.  It  must  suffice,  instead,  to  know 
that  the  connection  between  A  and  every  other  point  in  the 
orbit  is  rigid  and  exact.  It  is  quite  sufficient  for  present  pur- 
poses to  investigate  the  energy-fund  in  terms  of  the  conditions 
B  and  P  only,  as  bases. 

The  kinetic  energy  visible  at  B  in  the  velocity  U  is  divisible 
into  two  components,  the  radial  and  the  tangential,  respectively. 
To  each  of  these  must  correspond  a  respective  fund  of  energy, 
radial  anl  tangential  in  its  nature.  Now,  these  two  funds  of 
energy  bear  a  marked  contrast  with  each  other,  in  many  re- 
spects ;  and  as  this  contrast  runs  through  the  entire  question  of 
energetics,  it  is  worth  while  to  discuss  it  somewhat  carefully. 

In  the  first  place,  space-energy  can,  of  its  very  nature,  exist 
only  radially.  It  is  impossible  to  think  of  pure  separation  between 
two  bodies — which,  if  they  be  truly  single  bodies,  must  be 
regarded  as  geometric  points  at  their  centers — in  connection  with 
any  idea  of  direction  of  separation.  Moreover,  it  is  not  only 


36  ENERGY 

that  direction  of  separation  is  unthinkable  in  this  connection;  it 
would  have  no  effect  if  it  were  conceivable.  For  the  force  of 
gravitation  operates  equally  in  all  directions,  and  radially  only; 
and  as  space-energy  is  based  directly  upon  this  force,  it  too  must 
be  regardless  of  angular  direction. 

Now  this  energetic  force  of  gravitation  works  always  in  one 
direction,  of  the  two  directions  possible  within  each  radius.  It 
always  urges  the  two  bodies  together.  Urging  the  two  bodies 
apart,  at  all  times,  is  the  centrifugal  force;  and  this  force  is  a 
function  of  the  tangential  motion-energy.  The  two  forms  of 
energy,  spacial  and  kinetic,  are  therefore  in  energetic  counter- 
balance, just  as  much  as  their  forces  are  in  dynamic  counter- 
balance. When  the  space  of  separation  becomes  greater  than 
normal,  it  prevails  over  the  motion-energy  and  forces  the  two 
bodies  into  greater  propinquity.  When  the  motion  becomes 
greater  than  normal  —  and  this  is  always  the  result  of  the  action 
just  noted-  —  it  prevails  over  the  gravitational  attraction  and 
forces  the  bodies  apart.  Thus  these  two  forms  of  energy  vibrate 
against  each  other,  in  stable  equilibrium,  about  the  mean  ener- 
getic condition  as  a  center. 

Now  this  centrifugal  force,  although  itself  radial  in  direction, 
is  developed  only  by  tangential  motion;  and  it  is  in  permanent, 
not  vibratory,  equilibrium  that  the  tangential  component  of  the 
velocity  U  finds  itself  in  counterbalance  with  the  gravitational 
force.  To  explain,  let  us  consider  first  the  case  of  purely  cir- 
cular motion. 

In  this  case  the  radial  motion-energy  is  zero,  and  the  spacial 
separation  is  constant;  no  energy-transformation  whatever  takes 
place  and  no  energy  is  manifested  radially.  All  points  in  the 
circle  are  equally  mean  energetic  conditions.  The  balance  be- 
tween centripetal  and  centrifugal  forces  is  perfect  and  perma- 
nent. That  is  to  say,  speaking  mathematically, 

(centripetal)         (centrifugal) 
Radial  Force  -c  M.M-L.--  (16) 


c  U2 

D-  M7TMT  <17> 

In  the  second  member  of  Equation  16  the  reader  may  not  imme- 
diately  recognize  the  more  familiar  expression   for  centrifugal 


THE  MEAN  ENERGETIC  CONDITION  37 

force,    M-^-.     The   latter,   like   the   other   familiar   mechanical 
R 

equations  which  have  already  been  criticised,  is  a  special  approxi- 
mation, omitting  the  sum  of  the  masses;  which  is  sufficiently 
accurate  when  one  of  the  masses  is  so  large  that  variation  in  the 
other  does  not  appreciably  affect  their  sum. 

If,  now,  comparison  be  undertaken  between  two  circular 
motions  of  the  same  mass-pair,  one  at  a  radius  r  and  velocity  v 
and  the  other  at  radius  R  and  velocity  V,  the  change  in  space- 
energy  which  would  be  involved  in  a  passage  from  one  to  the 
other  would  be,  from  Equation  2, 

Ep-cM.JM-p—  -J-).  (18) 

The  change  in  velocity,  from  v  to  V,  which  must  occur 
simultaneously  in  order  that  centripetal  and  centrifugal  forces 
may  remain  balanced  and  the  orbit  remain  circular,  will  be  given 
by  Equation  17;  or 

-  (19) 


and  V2  =  (M,+M2)-.  (20) 

K. 

Therefore  the  change  in  kinetic  energy  involved  must  be 


But  the  last  term  of  this  equation  is  one-half  of  Equation  18, 
the  potential  energy  involved  in  passing  from  radius  R  to 
radius  r.  The  energy  absorbed  in  altering  the  velocity  is  there- 
fore one-half  the  amount,  and  of  opposite  sign,  from  that  re- 
leased in  increasing  the  propinquity  (if  the  second  radius  be 
considered  smaller  than  the  first).  The  net  energy  released, 
therefore,  is  the  algebraic  sum  of  the  two,  and  is  itself  equal  to 
the  last  term  of  Equation  21.  In  other  words,  when  masses  are 
brought  into  greater  propinquity  at  their  mean  energetic  condi- 
tion, or  with  circularity  conserved,  or  in  permanent  equilibrium 
between  centrifugal  and  centripetal  forces  (for  the  argument 
applies  equally  to  circular  motion  or  to  the  mean  energetic  point 
of  elliptical  or  hyperbolic  motion),  energy  must  be  abstracted; 
just  as  it  must  be  when  thev  are  allowed  to  fall  directly  together, 
vertically,  with  no  tangential  motion.  But  in  the  former  case  the 


38  ENERGY 

amount  of  energy  thus  to  be  abstracted  is  just  one-half  of  that 
requisite  in  the  latter  case. 

The  addition  to  the  circular  motion  of  a  radial  component, 
producing  elliptic  or  hyperbolic  orbit,  will  not  affect  this  universal 
equilibrium  between  tangential  motion  and  mean  energetic 
distance. 

In  the  case  of  free,  conic-section  motion,  on  the  other  hand, 
with  energy,  rather  than  circularity,  conserved,  no  energy  need 
be  abstracted  at  all,  as  the  propinquity  increases.  Indeed,  to 
attain,  under  a  fixed  mean  energetic  distance,  a  greater  pro- 
pinquity at  periastron,  energy  must  be  added,  and  in  such  a  way 
that  it  assumes  the  radial  form.  In  other  words,  to  produce  a 
permanent  consolidation  of  matter,  tending  to  continue  in  stable 
equilibrium,  energy  must  be  abstracted.  But  to  produce  a  tem- 
porary consolidation,  which  will  reconvert  itself  promptly  into 
separation  or  disgregation  of  matter,  energy  need  not  be  ab- 
stracted. It  even  needs  to  be  added. 

Further,  it  is  evident  from  Equation  17  that,  as  the  pro- 
pinquity increases  (or  the  distance  of  separation  decreases)  the 
velocity  must  increase.  Therefore,  putting  Equations  17  and  21 
together,  it  becomes  plain  that  when  masses  approach,  with 
circularity  conserved,  energy  must  be  abstracted  as  the  velocity 
increases — which  is  just  the  opposite  of  what  is  ordinarily  held 
to  be  a  universal  law.  But  in  the  case  stated,  it  is  to  be  remem- 
bered, it  is  not  the  radial,  but  purely  the  tangential,  velocity 
which  increases  as  energy  is  abstracted.  This  is  one  of  the  many 
ways  in  which  the  radial  and  tangential  forms  of  energy  are 
markedly  contrasted. 

Thus,  for  instance,  in  the  case  of  the  moon  and  the  earth, 
which  revolve  in  stable  equilibrium  and  in  an  orbit  which  is 
almost  circular,  a  mean  energetic  distance  of  about  240,000  miles 
is  maintained  permanently  by  a  mean  linear  speed  of  about  3100 
feet  per  second.  But,  if  the  earth  were  devoid  of  rotation  rela- 
tively to  the  sun,  the  friction  of  our  oceanic  tides  would  be  tend- 
ing steadily  to  slow  down  the  moon.  But  as  this  happens  the 
decreasing  centrifugal  force  between  moon  and  earth  becomes 
no  longer  able  to  counterbalance  the  gravitational  attraction  be- 
tween the  two;  and  moon  and  earth  tend  to  fall  together.  But 
as  this  occurs,  space-energy  is  released,  sufficient  not  only  to 
make  good  the  loss  of  energy  to  the  tides,  but  more.  The 


THE  MEAN  ENERGETIC  CONDITION  39 

equilibrium  remains  stable,  and  the  abstraction  of  tidal  energy 
has  the  apparently  paradoxical  effect  of  speeding  up  the  moon. 

As  a  matter  of  fact,  the  earth  is  revolving,  relatively  to  the 
sun,  in  the  same  direction  as  the  moon  moves  about  the  earth, 
and  with  a  higher  angular  velocity.  The  result  is  that  the  fric- 
tional  resistance  of  the  tides,  instead  of  tending  to  abstract 
energy  from  the  moon  for  the  speeding  up  of  the  earth,  tends 
to  slow  down  the  earth's  rotation,  with  the  transfer  of  energy 
to  the  moon.  In  this  case  the  moon  is  simultaneously  removed 
from  the  earth  and  retarded  in  tangential  linear  velocity. 

Either  case  illustrates  the  point,  viz :  that,  whereas  in  radial 
motion  the  energy  increases  directly  with  the  linear  velocity,  in 
tangential  motion  it  increases  inversely  therewith.  The  mere 
alteration  in  direction  from  radial  to  tangential,  which  has  been 
accomplished  by  gravitational  reaction  with  the  second  mass- 
portion,  has  constituted  a  reversal  of  algebraic  sign  of  the 
energy — algebraic  signs,  in  energetics,  being  significant  merely 
of  whether  the  energy  be  going  into  or  coming  out  of  a  given 
system,  or  of  departure  on  one  side  or  the  other  from  the  mean 
energetic  condition.  Although  it  is  universally  taught,  in  the 
engineering  schools,  that  energy  always  enters  matter  as  its 
velocity  increases,  and  vice  versa,  it  now  appears  that  this  is 
true  of  only  one-half  the  energy  of  the  universe — the  radial,  or 
perceptible,  half.  With  the  other  half — the  tangential,  or  latent, 
or  imperceptible,  half — velocities  increase  as  energy  departs  from 
matter,  and  vice  versa. 

This  innate  tendency  of  all  vibratory  forms  of  energy  period- 
ically to  alter  the  algebraic  sign,  with  every  transformation  from 
kinetic  to  potential,  or  from  radial  to  tangential,  or  the  reverse 
— as  visible  first  in  the  familiar  pendulum,  and  now  again  in  the 
action  of  any  free  mass-pair — is  a  fundamental  characteristic 
of  energy  which  is  of  the  utmost  importance.  Nature  knows 
nothing  of  smooth,  continuous  progress.  All  goes  by  pendulum- 
swings,  reversals  and  transformations,  as  a  ship  tacks  in  beating 
against  the  wind. 

It  is  in  such  terms  as  these  that  the  energy-fund  of  the  pair 
must  be  thought  of,  as  consisting  of  tangential  energy  plus 
radial  energy.  The  amount  of  tangential  energy  within  the  pair 
determines  the  mean  energetic  distance  D.  and  the  tangential 
component  of  the  mean  energetic  velocity  U.  It  remains  perma- 


40  ENERGY 

nently  fixed  in  form  and  quantity,  so  long  as  energy  is  neither 
added  nor  abstracted  from  without.  The  amount  of  radial 
energy  within  the  pair  determines  the  eccentricity  of  orbit  and 
the  radial  component  of  the  mean  energetic  velocity.  It  vibrates 
at  each  revolution  from  kinetic  to  potential  form  and  back. 

The  Radial  Energy-fund.  The  mathematical  expression 
for  the  fund  of  radial  energy  within  the  pair  may  be  had  from 
Equation  7.  By  algebraic  transformations  which  need  not  be 
reproduced  here,  it  may  be  stated  in  terms  of  either  the  mean 
energetic  or  the  periastron  condition.  It  may  also  be  expressed 
in  terms  of  either  space  or  velocity,  for  the  energy  alternately 
takes  either  form.  It  will  also  be  of  convenience  for  later 
purposes  if  the  kinetic  expression  be  stated  in  three  different 
ways.  Thus, 

(potential)  (kinetic) 

Radial  Energy  =  Er  =  2  c  MjM. 

(kinetic)  %  (kinetic) 

~,r,,    U2  cos2   a     1       rt     ,,„„   U2  sin2  a 


'*     M.  +  M,  '' 

Of  these  three  kinetic  expressions  the  first  and  simplest  will  be 
the  most  commonly  used.  Of  the  others,  the  last  two  will  some~ 
times  be  convenient  because,  in  the  mean  energetic  condition. 
U  cos  a  is  the  radial,  and  U  sin  a  the  tangential,  component  of 
the  velocity  U. 

Stated  in  terms  of  the  extreme  energetic  condition  at  perias- 
tron, S0  being  the  periastron  or  minimum  distance  of  separation 
and  V  the  periastron  or  maximum  velocity  of  motion,  Equation 
22  becomes 


(23) 

(It  is  to  be  noticed,  in  order  to  avoid  confusion  of  thought,  that 
at  periastron,  although  the  motion  is  there  purely  tangential  in 
direction,  radial  energy  nevertheless  is  present;  because  the  tan- 
gential velocity  is  there  so  great  that  the  radial  forces  are  un- 
balanced. The  radial  energy  takes  this  tangential  form  for  an 
instant  only.) 

Of  all  the  expressions  for  the  radial  energy  of  a  pair,  that 


THE  MEAN  ENERGETIC  CONDITION  41 

given  in  Equation  2,  in  potential  form,  is  the  first  to  get  well  in 
mind.  Equation  2  reads: 

Ep  =  cM1M2(-^-  .-L).  (2) 

As  the  greater  distance  of  separation  S  becomes  very  great,  its 
influence  upon  the  value  of  Ep  becomes  very  slight;  until,  at  the 
limit,  when  S  has  become  of  celestial  dimensions  between  bodies 
of  earthly  magnitude,  its  value  may  be  neglected.  Equation  2 

then  becomes  proportional  to  -^-  only.    The  same  is  true  of  the 

S0 
potential  forms  of  Equations  22  and  23,  wherein  the  space-factor 

appears  only  as  —  or  — .     For  this  reason  it  seems  convenient 
D         S0 

to  assign  to  this  reciprocal  of  radial  space  of  separation  the  term 
propinquity;  whereupon  it  may  be  said  that,  for  all  mass-pairs 
not  already  in  unusual  propinquity,  their  potential  energy  given 
out  is  proportional  to  their  propinquity.  And  in  any  event  it 
is  true  that  differences  in  potential  energy  are  proportional  to 
differences  in  propinquity. 

From  this  it  becomes  obvious  that  the  amount  of  energy 
which  must  be  abstracted  from  any  mass-pair  before  they  can 
be  brought  into  coincidence,  with  S0  =  o,  is  infinite.  Of  course, 
no  two  bodies  can  ever  be  brought  into  coincidence.  But  since 
the  obstacles  to  the  feat  lie  only  in  the  solid  dimensions  and  the 
density  of  the  two  masses,  which  are  variables  to  which  no  rule 
can  be  applied,  it  still  remains  true  that  the  amount  of  energy  in 
a  pair  which  lies  awaiting  abstraction  is  indefinite  in  amount. 
It  is  only  the  ability  to  abstract  it  which  is  limited. 

The  Tangential  Energy-fund.  When  attempt  is  made  to 
give  exact  mathematical  expression  to  the  tangential  fund  of 
energy,  trouble  arises.  In  tangential  motion  no  force  is  over- 
come by  that  motion,  as  is  the  case  in  radial  energy.  There  is 
no  transformation  of  energy.  Indeed,  there  is  not  even  any 
manifestation  of  energy.  When  careful  thought  is  taken  it 
appears  as  one  of  the  fundamental  principles  in  nature  that 
only  radial  energy  can  be  perceived  by  the  human  senses,  or 
by  any  other  external  mass-system.  A  revolving  pair,  unless 
it  be  of  such  dimensions  and  velocity  that  its  members  can  be 
perceived  separately,  as  they  alternately  approach  and  depart 
from  us,  does  not  appear  to  us  as  a  pair  at  all,  but  as  two  units. 


42  ENERGY 

Thus,  the  heavenly  bodies  are  all  of  such  dimensions  that  they 
can  be  perceived  separately,  and  from  them  we  get  our  first 
exact  ideas  concerning  tangential  energies.  Similarly,  such  re- 
volving pairs  as  fly-ball  governors,  fly-wheels,  etc.,  usually  re- 
volve at  a  slow  enough  speed  so  that  we  can  perceive  their  com- 
ponent parts;  and  so  we  learn  to  apply  these  exact  ideas  con- 
cerning tangential  energies.  But  if  the  fly-wheel  or  the  like 
revolves  so  rapidly,  or  becomes  so  minute,  that  its  component 
parts  are  no  longer  distinguishable,  then  we  can  perceive  its 
energy  only  when  we  come  into  contact  with  it  as  a  whole.  We 
then  learn— if  it  does  not  contribute  so  much  energy  to  us  that 
our  wits  are  at  fault — that  we  have  perceived  only  that  portion 
of  its  tangential  energy  which  has  ceased  to  be  tangential  and' 
become  radial. 

While  this  fact,  like  so  many  others  which  have  been  adduced 
to  the  present  argument,  is  of  comparatively  little  importance  in 
engineering  mechanics,  it  becomes  of  basic  importance  when  the 
revolving  pair,  or  system  of  pairs,  reduces  to  the  dimensions  of  a 
molecule  or  an  atom,  and  its  energy  becomes  known  to  us  as 
heat  or  the  like.  When  we  touch  a  body  and  perceive  that  it  is 
"hot"  we  are  like  some  vast  giant  who  might  be  too  big  to  see 
our  tiny  human  contrivances  on  the  surface  of  the  earth,  yet 
who  might  perceive  them  by  placing  the  tip  of  his  finger  upon  a 
field  in  which  were  many  rapidly  revolving  fly-wheels  or  buzz- 
saws.  He  could  not  see  that  each  fly-wheel  or  buzz-saw  was 
composed  of  many  parts,  balanced  in  their  motion  in  an  equilib- 
rium which  gives  them  the  appearance  of  unity ;  yet  to  our  senses 
it  would  be  an  obvious  truth. 

It  is  therefore  not  so  surprising  that,  since  we  have  no  power 
to  perceive  tangential  energy,  we  have  no  means  for  expressing 
it  mathematically.  For  it  is  a  fact  that  no  exact  statement  can 
be  made  as  to  the  tangential  energy-fund  existent  within  any 
pair.  Tangential  energy  is  there,  surely  enough ;  for  it  can 
come  out,  by  becoming  radial  in  form,  and  can  overcome  re- 
sistance. But  there  is  neither  any  definite  idea  nor  any  exact 
equation  for  the  total  amount  which  can  thus  come  out.  We 
know,  from  Equation  21,  that  the  amount  thus  available  is  one- 
half  the  radial  space-energy  available  from  any  given  same  change 
of  radial  separation.  But  as  the  latter  has  already  been  shown 
to  be,  so  far  as  exact  mathematical  limitations  appear,  infinite 


THE  MEAN  ENERGETIC  CONDITION  43 

in  amount,  the  only  statement  which  this  leads  us  to  is  that  the 
tangential  energy-fund  of  any  mass-pair  is  one-half  of  infinity; 
which  is,  of  course,  meaningless.  It  still  leaves  us  forced  to 
confess  that  for  the  tangential  fund  we  have  no  exact  expression. 

Our  knowledge  in  that  direction  is  much  like  that,  for  in- 
stance, of  a  man  who  owned  excellent  thermometers,  barometers, 
etc.,  all  of  which  had  slipped  their  scales.  He  would  know  ex- 
actly what  a  degree  of  alteration  of  temperature  was  like,  or  an 
inch  of  barometric  pressure.  He  could  report  usefully  to  his 
neighbors,  from  day  to  day,  upon  the  changes  in  the  weather. 
But  he  could  never  tell  them  exactly  how  hot  or  cold  it  was,  nor 
how  far  the  conditions  were  from  any  absolute  zero.  So  as  to 
tangential  energy,  we  can  measure  exactly,  in  joules,  foot-pounds 
or  what  you  please,  any  stated  change  in  tangential  energy,  from 
known  change  of  radius  of  purely  tangential  motion ;  but  beyond 
that  we  cannot  go.  We  cannot  determine  any  absolute  zero  or 
maximum  from  which  to  make  our  ideas  exact.* 

In  thinking  of  the  energy-fund  of  any  mass-pair  or  system, 
therfore,  all  idea  of  reducing  the  thing  to  any  exact  statement, 
founded  upon  an  absolute  zero,  must  be  abandoned  from  the 
start.  The  only  base  or  zero  which  has  become  visible  in  the 
preceding  discussion — which  has  been  just  as  exact  as  it  has 
been  possible  to  make  it,  short  of  dealing  with  infinitesimal 
masses — has  been  the  mean  energetic  condition.  It  is  only  in 
this  mean  energetic  condition  that  the  tangential  energy  is  directly 
visible.  It  is  the  amount  of  tangential  motion  on  hand  which 
determines  the  mean  energetic  distance  D,  or  the  space  occupied 
by  the  pair. 


*In  addition  to  the  above  it  must  be  noted,  before  leaving  this  contrast 
between  radial  and  tangential  energies,  that  the  radial  energy  alone  may 
be  considered  the  exclusive  property  of  the  mass-pair  itself.  It  has  already 
been  noted  that  between  a  pair  of  absolutely  single  bodies  only  radial 
separations  and  velocities  may  be  measured.  For  tangential  motions  the 
presence  of  at  least  three  mass-portions  is  necessary.  In  nature  this  is 
tantamount  to  saying  that  no  body  is  truly  a  single  body.  Every  natural 
body  possesses  dimensions ;  and  if  so,  tangential  motion  of  revolution  can 
be  perceived  by  comparing  its  various  portions.  It  is  thus  that  we  per- 
ceive the  motion  of  the  moon  around  the  earth ;  as  different  portions  of 
the  earth  swing  up  or  down  relatively  to  the  moon,  we  say  that  the  moon 
rises  or  sets.  But  to  those  telescopes  which  may  be  situated  upon  planets 
so  distant  that  the  earth  appears  as  a  point  of  light,  the  revolution  of  the 
moon  can  become  perceptible  only  by  a  comparison  with  distant  and 
apparently  fixed  stars ;  which  amounts  to  bringing  in  a  third  mass  very 
much  larger  than  any  of  those  yet  mentioned. 


44  ENERGY 

On  either  side  of  this  mean  energetic  condition  the  radial 
energy  of  every  energetic  pair  vibrates  as  does  a  pendulum  about 
its  supports.  But,  in  the  case  of  the  free  energy-system,  this 
support,  instead  of  being  fixed  and  definite,  is  floating  in  mid- 
space.  It  cannot  be  attached  to  or  measured  from  any  basis 
which  has  yet  been  devised.  The  most  which  can  be  done  is  to 
compare  it  with  other  mean  energetic  systems,  each  of  which  is 
equally  homeless.  As  the  sun  serves  as  a  central  basis  in  refer- 
ence to  which  the  motion  of  the  planets  may  be  conveniently 
measured,  so  the  mean  energetic  condintion  serves  as  a  central 
basis  or  zero-point  from  which,  in  either  direction,  the  radial 
energy  of  the  system  may  be  conveniently  measured.  But  both 
the  sun  and  the  mean  energetic  condition  float  indeterminately 
in  space,  without  any  possible  reference  to  any  absolute  base  or 
zero. 

Indeed,  the  prime  lesson  sought  to  be  imparted  by  this  paper 
is  that  there  is  not  anywhere  in  energetics,  in  any  department 
where  exact  knowledge  has  yet  penetrated,  any  reason  for  ever 
believing  in  the  existence  of  an  absolute  zero  for  anything.  So 
far  as  we  now  know,  there  is  no  absolute  zero  for  either  energy, 
velocity,  space,  force,  volume,  temperature  or  entropy;  nor,  so 
far  as  the  writer  is  aware,  for  the  similar  factors  of  those  other 
forms  of  energy  with  which  he  is  less  familiar,  such  as  chemical, 
electrical,  etc.  All  that  we  have  ever  perceived  in  any  of  them 
is  a  vibration  on  either  side  of  some  central  mean  value,  which 
itself,  in  turn,  cannot  be  regarded  as  fixed. 

While  this  topic  must  receive  further  analysis,  in  later  papers 
of  the  series,  before  it  can  be  understood,  it  should  be  an  attribute 
of  energy  familiar  to  every  student,  as  its  basic  characteristic, 
that  energetic  potentialities  consist  as  much  in  departures  upon 
one  side  of  the  central  mean  as  the  other.  For  instance,  in 
thinking  of  the  energy  embodied  in  mundane  gravitation,  such  as 
water-power,  it  should  be  remembered  that  very  low  conditions 
of  matter  embody  energy,  as  well  as  very  high  ones.  When 
one  wishes  to  purchase  water-power  it  is  usual  to  buy  some 
basin  which  nature  keeps  filled  with  water  at  an  appreciable 
elevation  above  the  sea-level — which  latter  is  in  this  case  the 
mean  energetic  level.  But  if  the  valley  of  the  Salton  Sea,  in 
California,  or  the  Caspian  or  Saharan  basins,  or  any  other  similai 
depressions  below  the  sea-level,  which  nature  would  keep  emptied 


THE  MEAN  ENERGETIC  CONDITION  45 

of  water  by  evaporation,  happened  to  be  conveniently  near  the 
sea  and  on  the  market,  they  would  constitute  invaluable  water- 
powers.  With  their  help  the  waters  of  the  ocean,  which  are 
commonly  regarded  as  having  reached  the  absolute  zero  of 
hydraulic  head,  would  become  gigantic  sources  of  hydraulic 
energy. 

Similarly  with  heat-engines,  it  is  only  by  chance  that  it 
happens  to  be  the  most  convenient  thing,  when  one  wishes 
power,  to  buy  coal,  fit  to  give  out  some  15,000  B.t.u.  per  pound 
at  a  temperature  some  hundreds  of  degrees  above  the  mean 
thermal  level  for  the  surface  of  the  earth.  On  the  other  hand, 
if  a  mine  should  be  discovered  from  which  could  be  procured  a 
durable  solid  which  would  bear  transportation,  and  which  would 
absorb,  instead  of  develop,  some  15,000  B.t.u.  per  pound  at  a 
temperature  even  two  hundred  degrees  below  that  thermal  mean 
(whereas  ice  will  absorb  only  about  140  B.t.u.  per  pound  at  a 
temperature  of  32°  Fahr.),  the  owner  could  sell  it  in  unlimited 
quantities  for  heat-engine  purposes,  for  chilling  the  condensers 
so  low  that  ordinary  sun-heat,  even  in  winter,  would  suffice  for 
the  motive  heat.  If  this  novel  substance  happened  to  be  more 
convenient  than  coal  for  any  reason,  all  our  better  heat-engines 
would  soon  be  designed  in  every  line  with  a  view  to  its  use,  just 
as  they  now  are  for  the  use  of  coal-made  steam,  or  gas,  or  oil. 

Xor  is  this  mere  phantasy.  Since  the  temperature  of  our 
condensers  is  just  as  far  below  that  of  our  steam-boilers  as  that 
of  the  boilers  is  above  the  condensers,  our  steam-engines  must 
be  regarded  as  evincing  the  availability  of  cold  for  work-per- 
formance as  much  as  that  of  heat  for  the  same  purpose.  If  any 
steam-engine  owner  does  not  believe  that  he  is  running  a  cold- 
engine,  let  him  allow  his  condenser  to  warm  up.  He  will  be  in 
exactly  the  same  trouble  as  if  he  allows  his  boiler  to  cool  off. 
And  as  a  matter  of  fact  it  is  quite  as  likely  that  there  exist 
somewhere  in  the  universe  deposits  of  substance  which  embody 
chemically  the  chill  of  interstellar  space  as  that  here  on  earth 
are  deposits  of  coal  which  embody  chemically  the  incandescence 
of  intrasolar  mass — though  they  are  not  likely  to  be  convenient 
solids,  like  coal. 

The  conclusions  which  have  been  sought  in  the  foregoing 
argument  may  be  summarized  briefly  as  follows : 


46  ENERGY 

1.  The  energy-fund  of  any  mass-pair  is  made  up  of  two 
contrasted  sorts,  viz :  the  radial  and  the  tangential. 

2.  Of  these,  only  the  radial  fund  is  capable  of  exact  mathe- 
matical expression ;  or,  so  far  as  the  argument  has  yet  proceeded, 
of  manifestation  to  any  external  mass-system. 

3.  The  variations  of  either  fund  occur,  not  toward  or  away 
from  any  known  absolute  zero  of  any  condition,  but  on  either 
side  of  a  mean  energetic  condition;   and   this  mean  energetic 
condition  itself  cannot  be  located  absolutely,  but  only  relatively 
to  other  mean  energetic  conditions. 

4.  The  tangential  energy-fund,  itself  unmeasurable,  acts  as 
mean  central  base  for  the  radial  fund,  on  either  side  of  which  the 
latter  vibrates.     For  the  tangential  fund  the  mean  central  base 
as   yet    remains   undefined;    although    it   is   obvious,    from    the 
generally  stable  equilibrium  of  energetic  phenomena,  that  such  a 
stable  central  condition  must  exist. 

5.  Of  the  total   energy-fund  of  any  mass-pair,   the   radial 
energy  constitutes  the  perceptible,  the  measurable  and  the  trans- 
missible portion.     It  is  the  medium  of  communication  between 
the  pair  and  the  outside  universe.    It  may  properly  be  styled  the 
sensible  energy.     In  its  potential   form  it  is  distinguishable  as 
either  an  unusual  degree  of  spacial  separation,  or  as  an  unusual 
lack  of   force    (as    at   the   apastron   of   the   illustrative    elliptic 
orbit).     In  its  kinetic   form   it  is   distinguishable  as   either   an 
unusual  degree  of  unbalanced  force,  or  as  an  unusual  lack  of 
spacial    separation    or   unusual    concentration    of    mass    (as    at 
periastron  of  the  illustrative  ellipse).    These  statements  will  be 
found  to  apply  broadly  to  the  thermal  and  mechanical  energies 
of  all  forms  of  matter,  as  well  as  to  the  elementary  illustration. 

6.  Of  the  total  energy-fund  of  any  mass-pair,  the  tangential 
energy   is   the    imperceptible,   the   immeasurable    and    the   non- 
transmissible  portion.     It  is  the  elastic  base  upon  which  rests 
the  radial  energy.    It  is  the  means  for  the  storage  of  perceptible 
energy  received  radially  from  without,  and  likewise  the  source 
from  which  is  drawn  the  energy  manifested  radially  to  the  out- 
side world.     It  may  properly  be  styled  the  latent  or  invisible 
energy.     It  is  not  to  be  perceived  directly  at  all ;  and  indirectly 
it  is   to  be   distinguished  solely  by  its   ability   to  turn  into   or 
absorb  radial  energy. 

7.  All  energetic  measurements  must  be  made  from,  and  all 


THE  MEAN  ENERGETIC  CONDITION  47 

concepts  based  upon,  not  any  absolute  zero  of  anything,  which 
exists  as  a  fixed  support,  but  from  a  central,  mean-energetic 
condition,  which  itself  floats  unsupported  in  mid-space,  like  our 
sun  in  the  heavens.  While  the  position  of  this  mean-energetic 
condition  must  itself  be  subject  to  natural  law,  yet  it  is  con- 
trolled by  forces  and  phenomena  too  large  to  be  taken  into 
consideration  in  any  concrete  case. 

8.  Every  mass-pair  embodies  some  radial  energy  and  some 
fund  of  tangential  energy.  This  fact  has  not  yet  been  developed 
in  the  argument,  but  it  should  be  stated  here,  in  company  with 
the  preceding  seven  principles.  When  the  eccentricity  of  the 
mutual  orbit  of  the  pair  is  great,  the  radial  fund  is  large  in 
proportion  to  the  tangential.  When  the  eccentricity  is  small,  the 
radial  fund  is  small.  But  as  the  eccentricity  can  never  be  con- 
ceived as  becoming  either  zero  or  infinity,  there  must  always  be 
some  finite  fund  of  radial  energy.  And  since  the  pair  can  never 
be  conceived  as  united  into  complete  coincidence,  but  must  always 
occupy  some  space,  there  must  always  be  some  fund  of  tangential 
energy.  This  idea  will  be  developed  later. 


CHAPTER  IV. 

THE  Two  FACTORS  OR  DIMENSIONS  OF  ENERGY. 

In  all  of  the  mathematical  expressions  for  energy  which  have 
been  developed  in  the  preceding  chapters  there  have  everywhere 
appeared  two  factors.  One  of  the  factors  is  the  product  M^Ma 
of  the  separate  masses  involved.  The  other  is  some  function  of 
either  the  space  or  the  motion  involved  in  their  separation  into 
duality.  It  is  obvious  that  these  two  factors  possess  a  distinct 
and  contrasted  significance. 

The  first  of  these  factors,  MjM2,  is  a  measure  of  the  extent 
to  which  duality  exists,  in  which  may  be  embodied  the  space  or 
motion  relationships  which  are  measured  by  the  other  factor.  It 
is  a  measure  of  the  extent  of  "mass-pairing,"  as  we  shall  call  it 
for  convenience,  which  is  present.  It  has  therefore  been  named 
by  the  writer  the  "extent"  of  energy,  or  the  "extensity,"  present. 

The  second  factor  gives  the  degree  of  space  or  motion  em- 
bodied within  the  mass-pair;  and  as  it  appears  to  human  senses 
as  the  feature  of  energy  which  gives  evidence  of  the  degree  of 
concentration  of  energy,  by  the  sharpness  of  sensation  or  other 
effect  which  the  energy  may  produce,  it  has  been  called  by  the 
writer  the  "intensity"  of  energy  present.  Of  the  two  factors 
this  latter  is  the  more  familiar  to  students  of  energetic  phe- 
nomena, and  will  therefore  be  considered  first. 

The  Intensity  of  Energy.  If  the  fundamental  expressions 
for  potential  and  kinetic  energy  be  divided  by  the  factor  M1M2, 
so  as  to  derive  mathematical  expressions  for  the  intensities  of 
these  two  energies,  there  results,  for  the  potential  intensity, 
from  Equations  2  and  22, 


and  for  the  kinetic  intensity,  from  Equations  5  and  22, 

1       V2_Vo2     _o  U2  e 

lk      2'  '    J  2 


From  these  equations  the  intensity  of  energy  appears  broadly  as 

48 


THE  TWO  FACTORS  49 

a  function  of  (i)  for  potential  intensity  a  difference  in  "pro- 
pinquities/' or  reciprocals  of  spacial  separation;  and  (2)  for 
kinetic  intensity  a  similar  difference  in  ratio  of  velocities-squared 
to  aggregate  mass  involved. 

In  kinetic  mechanical  engineering,  it  is  the  prime  charac- 
teristic of  the  intensity  of  energy  that  it  controls  the  direction 
of  energy-transformations.  It  is  a  fact  familiar  to  engineers 
that  it  is  the  body  possessing  the  higher  velocity,  even  if  the 
smaller  mass,  which  overtakes,  collides  with  and  gives  its  energy 
to  the  more  slowly  moving  masses.  Our  entire  experience  with 
hammers,  projectiles,  and  railway-collisions  is  but  experience 
with  the  intensity  of  kinetic  energy,  and  its  promotion  of  energy 
transformation. 

But  now  it  appears  that,  whereas  our  present  sub-conscious 
idea  of  kinetic  intensity  attaches  itself  merely  to  velocity,  the 
exact  affair  is  in  reality  the  ratio  of  velocity-squared  to  aggregate 
mass.  Like  all  other  energetic  phenomena,  its  manifestation 
here  upon  the  earth's  surface,  with  the  earth  for  one  of  the 
participating  masses  in  every  pair,  has  been  disguised  by  the 
fact  that  here  the  aggregate  me.ss,  viz:  projectile  plus  earth, 
remains  virtually  constant,  and  so  has  been  dropped  from  con- 
sideration of  the  variables.  But  when  the  bodies  concerned  are 
of  molecular  magnitudes,  with  nothing  known  as  to  comparative 
masses  of  the  members  of  any  pair,  it  is  obvious  that  the  de- 
nominator of  the  ratio  mav  become  as  active  a  factor  as  the 
numerator,  in  the  variation  of  intensity. 

As  to  the  potential  intensity  of  energy,  that  is  also  a  familiar 
fact  in  every-day  engineering;  but  not  quite  in  the  form  pre- 
sented here. 

In  the  first  place,  potential  intensity,  or  intensity  of  energy 
due  to  an  unusual  departure  of  spacial  separation  from  the 
average,  may  assume  either  of  two  forms,  viz :  unusual  separa- 
tion, on  the  one  hand,  or  unusual  lack  of  separation  on  the  other. 
That  is  to  say,  potential  energy  may  be  observed  as  great  either 
because  S  is  very  much  greater  than  the  mean  energetic  distance 
D,  or  because  S0  is  very  much  smaller. 

All  of  the  cases  in  engineering  which  are  commonly  recog- 
nized as  potential  mechanical  energy,  such  as  suspended  weights, 
mill-pond  water,  or  projectiles  at  the  summit  of  their  trajectories, 
belong  in  the  former  class.  They  are  in  reality  following  the 


50  ENERGY 

apastrons  of  extremely  eccentric  orbits,  which  pass  closely  about 
the  center  of  the  earth — which  orbits,  of  course,  can  never  be 
completed  because  of  the  solid  obstruction  of  the  earth's  body. 
Their  mean  energetic  distances  are  a  few  miles  at  most,  perhaps 
a  few  feet  only,  from  the  earth's  center;  whereas  their  separa- 
tions from  that  center  while  we  are  using  them  is  some  four 
thousand  miles,  the  earth's  radius.  Moreover,  they  possess  very 
little  tangential  motion  parallel  with  the  earth's  surface.  There- 
fore, their  gravitational  force  is  almost  entirely  unbalanced.  We 

consider  _5i_ ,  in  Equation  24,  to  be  a  constant,  and  S0  in  the 
SS0 

parenthesis  to  be  negligible.  The  distance  of  separation  S — or 
h,  for  height  or  head,  as  it  is  commonly  called — remains  alone, 
ready  for  us  to  consume  of  it  what  small  fraction  we  may. 

But  we  also  know  potential  intensity,  in  engineering,  as  an 
unusual  lack  of  separation  in  mass.  As  we  compress  an  elastic 
gas,  or  a  steel  spring,  for  instance,  into  a  volume  less  than  its 
condition  of  mean  energetic  equilibrium,  there  is  developed  a 
latent  potentiality  for  energy  which  manifests  itself  as  an  unusual 
intensity  of  -force.  This  force  does  not  appear  to  the  average 
engineer,  it  is  true,  as  an  integration  of  the  many  unbalanced 
centrifugal  forces  of  a  multitude  of  tiny  orbits  at  periastron. 
But  there  is  no  other  unbalanced  force  in  the  elements  of 
mechanics  which  so  fits  the  conditions  as  to  be  acceptable  as  an 
explanation  of  it. 


Force,  it  is  also  true,  is  a  function  not  only  of  -i-  (or 


i 


S         volume 

as  we  should  state  it  for  gases),  but  also  of  the  masses  involved. 
Therefore  it  is  not  a  pure  manifestation  of  intensity.  But  when 
the  mass-factor  happens  to  be  constant,  as  is  commonly  true  in 
engineering  mechanics  and  as  may  sometimes  be  true  even  in 
molecular  mechanics,  it  becomes  a  true  measure  of  intensity. 

In  this  case,  also,  the  characteristic  of  intensity  as  the  de- 
terminator  of  the  direction  of  energy-transformation  appears. 
It  is  always  the  greater  force  which  overcomes  and  contributes 
energy  to  the  smaller  force.  It  is  always  the  smaller  force  which 
receives  and  stores  it.  That  is  why  our  fundamental  concept  of 
energy  covers  the  action  of  all  the  unbalanced  force  present, 
acting  through  an  infinitesimal  distance,  rather  than  all  the  dis- 
tance covered,  acted  upon  by  an  infinitesimal  element  of  the  force. 


THE  TWO  FACTORS  51 

That  is  to  say,  the  mathematical  equation  for  this  fundamental 
concept  of  an  infinitesimal  element  of  energy  is  Force  X 
d (Space),  and  not  Space  X  d (Force). 

Therefore,  while  the  mathematical  equations  are  the  only  true 
guides,  yet  the  general  concept  of  potential  intensity  may  be  based 
upon  either  force  or  space,  force  being  understood  to  be  synony- 
mous with  lack  of  space,  or  reciprocal  of  space.  When  the  in- 
tensity is  in  the  form  of  unusual  space  between  the  mass- 
portions,  the  unbalanced  force  is  centripetal,  gravitational  or 
concentrative,  and  is  comparatively  slight.  When  the  intensity 
is  in  the  form  of  unusual  lack  of  space,  the  unbalanced  force  is 
centrifugal,  or  disgregative,  and  is  very  great.  The  first  offers 
us  much  distance  traversible  with  little  force;  the  latter  makes 
available  a  greater  force  operative  through  a  smaller  distance. 
The  first  typifies  the  action  of  suspended  weights ;  the  latter  that 
of  compressed  elastic  matter. 

Thus,  either  unusual  attenuation  or  unusual  condensation  of 
matter  constitutes  potential  energy.  Unfortunately,  at  present 
we  quite  lack  suitable  names  for  these  two  contrasted  types  of 
potential  energy.  If  we  were  to  coin  words  for  them,  disgregic 
and  congrcgic  energies  would  be  the  natural  names.  But  this  is 
merely  a  suggestion. 

In  the  above  statements  all  words  of  degree,  such  as  great 
or  little,  are  used  only  in  a  comparative  sense,  for  any  given 
mass-system.  For  in  natural  mechanics  all  human  standards  of 
dimension  disappear.  The  mass-action  of  solar  systems,  base- 
balls and  molecules  is  now  believed  to  be  the  same,  in  principle. 
Any  such  a  system  may  be  taken  as  a  standard  of  dimension; 
but  the  ideas  of  great  or  small  applied  from  these  different 
bases  must  be  understood  as  fitting  widely  different  experiences. 

The  Extensity  of  Energy,  or  Degree  of  Mass-pairing. 
The  extensity-factor  of  energy  M±M2  appears  always  as  a 
product  of  two  separate  masses,  rather  than  as  their  sum.  This 
is  so  because  energy  can  exist  only  between  masses,  and  not 
throughout  mass.  That  is  to  say,  the  symbols  Mj  and  M2  signify 
that  within  the  first  body  there  reside  M±  units  of  mass,  while  in 
the  other  body  reside  M2  units.  Any  single  mass-unit  in  Mt 
would  then  form,  with  the  several  single  units  in  the  other  body, 
and  across  the  gap  between  the  two  bodies,  M2  unit  mass-pairs. 
The  entire  community  of  mass-units  in  Mt  would  therefore  form 


52  ENERGY 

M,.  X  M2  unit  mass-pairs  across  the  gap  separating  the  two 
bodies. 

Remembering  that  energy  can  exist  only  in  the  gap  between 
separate  mass-portions,  and  never  in  the  mass  itself,  it  becomes 
clear  that  the  factor  M1M2  is  the  extent  of  mass-pairing  which 
is  energetically  active  across  this  gap.  It  is  clear,  too,  that  in 
any  energetic  system  the  total  extent  of  mass-pairing  present,  or 
2  MM,  in  which  to  embody  intensities  of  relationship,  is  just  as 
much  a  factor,  and  may  be  just  as  variable  a  factor,  in  the 
energy  as  is  the  intensity  of  relative  space  or  motion  itself.  It 
is  because  we  have  so  long  been  accustomed,  in  our  engineering, 
to  splitting  off  from  the  earth  the  mass  of  a  railroad-train  or  a 
cannon-ball  and  treating  only  of  its  energy  relatively  to  the  earth 
— in  which  case  the  mass-pairing  factor  remains  constant  so 
long  as  the  separation  between  earth  and  machine,  or  the  energy 
itself,  lasts — that  it  has  been  forgotten  that  the  mass-pairing 
factor  is  itself  as  likely  to  experience  variation  as  is  the  intensity- 
factor  of  relative  space  or  motion.  And  in  molecular  mechanics 
it  is  perhaps  even  more  likely. 

This  variation  in  extent  of  mass-pairing  may  take  place  in 
either  of  two  ways.  First,  the  mass  of  either  party  to  the  energy 
may  increase  or  diminish ;  in  which  case  Mt  -f-  M.,,  or  2  M, 
varies  with  2  MM.  This,  according  to  the  ordinary  teachings 
of  mechanics,  is  what  always  occurs.  We  are  taught,  in  fact, 
that  we  can  increase  the  energy  between  cannon-ball  and  cannon 
(for  a  fixed  muzzle-velocity)  only  by  increasing  the  mass  of 
both  cannon  and  projectile. 

But  there  is  a  second  method  of  increasing  the  massivity  of 
energy  present,  and  that  is  by  increasing  the  number  of  cannon 
and  projectiles.  At  first  glance,  this  seems  to  be  the  same  as 
before,  viz :  energy  increasing  in  proportion  with  mass.  The 
difference  appears  when  it  is  remembered  that  the  earth  is  a 
party  to  almost  all  engineering  energies,  and  that  their  true 
nature  comes  out  only  when  matters  are  expanded  to  a  scale 
commensurate  with  the  dimensions  of  the  earth.  The  earth 
weighs  about  6  X  io21  long  tons.  A  cannon  and  projectile 
which  together  weighed  this  amount,  and  yet  possessed  only  the 
muzzle-velocity  of  standard  cannon,  would  be  a  very  mild  affair, 
as  planetary  energies  go.  But  compare  with  this  6  X  io21  long 
tons  of  standard  cannons  and  projectiles,  all  going  at  once,  and 


THE  TWO  FACTORS  53 

each  of  the  projectiles  equipped  with  energy  relatively  to  the 
entire  remaining  mass — and  some  concept  may  be  had  of  the 
energetic  possibilities  of  mass-subdivision!  In  comparing  these 
two  cases,  the  2  M,  or  aggregate  mass,  of  the  two  are  the  same; 
but  in  the  latter  case  the  2  MM,  or  extent  of  mass-pairing,  is 
very  much  the  greater. 

The  distinction  involved  here  is  hard  to  bring  out  clearly,  for 
man  has  had  no  experience  with  mass-systems  consisting  of  an 
earth's- weight  of  cannon,  all  going  off  at  once.  Yet  this  is  the 
only  true  mechanical  simile,  in  comparison  with  the  placid  old 
moon-earth  system,  to  a  white-hot  cannon  in  comparison  with 
the  same  cannon  cold  and  solid,  discharging  a  single  shot.  It  is 
because  human  experience  covers  no  gradations  between  these 
two  extremes,  of  matter  acting  as  a  solid  unit  on  the  one  hand 
or  as  an  innumerable  multitude  of  separate  solid  units  on  the 
other,  that  we  have  always  regarded  "work"  as  one  thing  and 
"heat''  as  a  very  different  thing — that  we  have  arranged  all  our 
formulae  for  mechanical  energy  with  the  mass-factor  appearing 
as  M  instead  of  as  M2,  neglecting  entirely  the  wide  range  of 
variation  of  which  the  factor  2  MM  is  capable  while  2  M 
remains  constant. 

Yet  this  distinction  is  all-important,  as  lying  at  the  heart  of 
the  comprehension  of  any  of  the  more  obscure  forms  of  energy, 
such  as  heat.  In  order  to  grasp  it,  it  must  be  remembered  how 
necessary  has  been  the  common  habit  of  speaking,  even  in 
celestial  mechanics,  of  the  bodies  involved  as  if  they  were  solid 
homogeneous  units.  In  discussing  the  energy  embodied  between 
moon  and  earth,  for  instance,  it  has  been  customary  to  consider 
both  moon  and  earth  as  perfectly  solid,  unit  spheres,  each  acting 
as  if  concentrated  at  its  own  center.  Yet  this  is  very  far  from 
the  truth.  Each  of  these  "units"  is  in  reality  extensively  and 
minutely  subdivided  into  interacting  energetic  mass-pairs.  We 
who  inhabit  the  earth  know  full  well  of  that  planet,  if  not  also 
of  the  moon,  that  every  energy  which  forms  the  subject  of  a 
human  science,  in  mechanics,  hydraulics,  meteorology,  chemistry, 
biology  and  electricity,  stands  in  evidence  of  minute  subdivision 
of  mass.  The  earth,  instead  of  being  a  homogeneous  unit,  is  in 
reality  subdivided  into  a  vast  and  most  intricately  organized  host 
of  parts,  of  differing  densities  and  other  characteristics;  and 
between  these  divided,  distinct  and  independent  particles,  of 


54  ENERGY 

rock,  water,  air  and  the  various  chemical  elements,  exist  the 
most  diverse  funds  of  energy,  both  potential  and  kinetic.  Merely 
to  mention  the  unquestionably  mechanical  ones,  there  are  the 
energies  of  the  winds,  waves,  tides  and  waterfalls,  all  the  pon- 
derous machines  of  human  construction,  and  the  innumerable 
host  of  moving  arms,  legs,  wings,  fins  and  claws  besides.  There 
is  not  only  energy  in  the  abysmal  gap  between  moon  and  earth, 
but  there  is  energy  in  each  of  the  myriad  of  tiny  crevices  in  each. 
Wherever  exists  subdivision  there  exists  energy. 

Indeed,  all  that  is  meant,  in  speaking  of  the  moon  and  earth 
or  any  other  body  as  a  "solid"  unit  of  mass,  is  that  for  the  par- 
ticular purposes  of  this  particular  sort  of  energy  the  mass- 
portions  concerned  may  be  considered  as  solid  units;  that  is, 
acting  as  if  their  mass  were  concentrated  at  a  single  point.  So 
long  as  this  special  assumption  holds  good,  2  MM  must  be 
regarded  as  a  constant,  as  well  as  S  M,  and  the  amount  of  this 
particular  sort  of  energy  is  proportional  to  the  first  power  of 
the  mass  involved.  But,  considering  all  sorts  of  energy  together, 
we  know  of  no  instance  in  which  this  special  assumption  holds 
true  broadly.  Human  experience  has  never  yet  encountered  a 
homogeneous  or  true  solid.  Like  the  geometric  point  or  line, 
the  solid  state  of  matter  is  merely  a  convenient  figment  of  the 
human  brain. 

It  has  been  altogether  natural,  and  even  necessary,  in  the 
study  of  applied  mechanics,  to  consider  mass  thus,  as  lumped 
into  portions  which  were  solid  units,  acting  perfectly  in  unison. 
But  for  a  mechanical  concept  of  heat  such  an  idea  is  fatal;  and 
even  for  the  purposes  of  applied  mechanics  it  seems  to  the 
writer  to  have  been  overdone.  For  it  is  essential,  for  the  com- 
prehension of  any  of  the  several  other  energies  which  are  daily 
growing  in  importance,  to  have  constantly  in  the  mind's  eye  a 
picture  of  mass,  not  as  a  homogeneous  unit,  or  solid,  but  as  a 
heterogeneous  system,  subdivisible  again  and  again,  into  smaller 
and  smaller  portions,  just  as  far  as  human  perception  is  able 
to  penetrate — into  molecules,  atoms,  ions  and  what  not.  The 
history  of  science  justifies  no  other  view.  Each  new  stage  of 
scientific  progress  has  revealed  some  further  refinement  of  sub- 
division of  matter  which  had  previously  escaped  the  more  clumsy 
perceptions  of  cruder  times. 

The  difference  to  the  comprehension  of  the  general  nature  of 


THE  TWO  FACTORS  55 

energy  lies  in  the  fact  that  if  the  homogeneous  dogma  be  true, 
the  purely  mechanical  energy  which  may  be  embodied  in  any 
given  mass  is  limited  by  the  velocity  or  distance  relatively  to 
some  fixed  base,  such  as  the  sun,  which  may  be  imparted  to  it; 
in  short,  to  a  variation  in  the  intensity-factor  of  its  energy. 
Thus,  in  comparing  the  energetic  possibilities  of  different  masses, 
these  must  vary  directly  as  the  mass.  But  if  the  heterogeneous 
dogma,  based  upon  the  equations  given  in  these  papers,  be  prop- 
erly understood,  then  mass  appears  as  capable  of  embodying 
within  itself  any  amount  of  energy  whatever,  not  only  by  varia- 
tions in  its  intensity  of  energy,  but  also  by  variations  in  its 
extensity  of  subdivision,  or  comminution. 

This  can  perhaps  be  made  clearer  by  an  illustration. 

Let  us  consider  a  form  of  energy  which  we  may  call  military 
or  combative  energy.  Suppose  there  exists  an  army  of  twenty 
thousand  men.  If  one  of  these  men  should  desert  his  duty,  the 
power  of  the  remaining  nineteen  thousand  and  odd  arrests  him 
— perhaps  only  after  some  desperately  violent  "intensity"  of 
resistance,  the  utmost  combative  energy  that  any  one  man  might 
arouse  in  the  face  of  superior  numbers.  This,  in  the  particular 
illustration  chosen,  is  the  minimum  possible  degree  of  sub- 
division of  the  army's  mass  into  combatively  opposed  portions. 
The  extensity  of  combative  energy  present  is  given  by  multiplying 
one  by  19,999 ;  while  the  intensity  of  combative  energy  depends 
upon  the  strength  and  spirit  of  the  individual  thus  split  off  from 
the  corps. 

The  portion  of  the  army  thus  split  off,  in  disobedience,  of 
course  might  be  greater  than  a  single  man.  The  number  in 
revolt  might  grow  most  gradually,  man  by  man,  until  it  had 
become  full  half  the  army.  If  we  suppose,  for  simplicity,  that 
each  man's  intensity  of  combative  ability  were  equal  to  that  of 
his  fellows,  then  the  energy  of  combat  would  grow,  as  the 
revolt  spread,  proportionately  to  the  product  of  the  numbers 
engaged  upon  the  two  sides.  And  this  would  reach  a  maximum, 
at  10,000  X  10,000  =  100,000,000,  when  the  opposed  parties  had 
become  equal. 

But  this  is  not  the  maximum  extensity  of  combative  energy 
of  which  the  army  is  capable.  It  is  merely  the  maximum  attain- 
able when  only  a  single  subdivision,  into  only  two  portions, 
occurs.  Suppose,  however,  that  by  the  time  the  revolt  had 


56  ENERGY 

absorbed  one-half  of  the  army  it  became  civil  war,  one  side 
adopting  a  red  uniform  and  the  other  a  blue.  Suppose,  then, 
that  the  spirit  of  disaffection  so  spread  that  further  subdivision 
took  place  within  each  faction.  The  red  army  splits  into  a 
"yellow"  and  a  "gray,"  of  five  thousand  men  each,  while  the 
blue  army  splits  similarly  into  a  "green"  and  a  "purple."  Each 
of  these  new  armies  raises  a  banner  of  its  own,  and  opposes  any 
and  all  comers.  What  is  now  the  possible  number  of  individual 
personal  conflicts?  What  is  now  the  extensity  of  military  or 
combative  energy  ? 

Between  the  four  armies  are  possible  six  different  battle- 
arrays — a  yellow-green,  a  yellow-gray,  a  yellow-purple,  a  gray- 
green,  a  gray-purple  and  a  green-purple.  But  the  possible  num- 
ber of  personal  conflicts  in  each  battle  has  now  been  reduced  to 
only  5000X5000  =  25,000,000,  or  one-quarter  as  great  as 
before.  The  aggregate  number  for  all  the  armies,  together,  there- 
fore, is  but  6  X  25,000,000  =  150,000,000.  This  measure  of  the 
extensity  of  combative  energy  proves  to  be  only  fifty  per  cent, 
greater  than  when  the  army  was  divided  into  only  two  equal 
portions. 

But,  obviously,  the  extent  to  which  the  army  may  subdivide 
into  equal  portions,  in  mutual  discord,  is  not  limited  to  four  such 
parts.  The  possibilities  in  this  direction  are  not  exhausted  until 
each  individual  soldier  has  become  a  knight  errant,  under  his 
own  standard,  and  the  field  a  proverbial  Donnybrook  Fair.  The 
extensity  of  combative  energy  would  then  be  much  greater  than 
150,000,000,  it  is  clear;  and  yet  it  is  equally  clear  that  it  would 
not  have  grown  in  proportion  to  the  number  of  subdivisions. 

Also,  it  is  to  be  noted  before  passing  to  mathematical  exact- 
ness in  the  discussion,  the  degree  of  subdivision  of  the  army  is 
limited  to  the  degrees  specified,  only  because  the  discussion  has 
been  confined  to  military  energy,  a  form  in  which  the  unit  mass- 
factor  is  a  single  soldier-pair.  That  is  to  say,  for  this  special 
purpose,  it  has  been  assumed  that  each  soldier  is  a  solid,  homo- 
geneous, indivisible  unit,  containing  no  internal  subdivisions  and 
energies.  This,  of  course,  is  not  true  in  speaking  of  energies  in 
general;  for  each  soldier  is  a  most  intricate  conglomeration  of 
separate  organs,  muscles,  glands,  bones,  cells,  etc.,  and  is  capable 
of  embodying  much  physiological  energy  even  in  times  of  peace, 
when  the  army  remains  a  solid  unit.  Only,  for  the  particular 


THE  TWO  FACTORS  57 

purposes  of  combative  energy  the  unit  mass-pair  is  a  couple  of 
soldiers,  completely  equipped ;  just  as  for  the  particular  purposes 
of  heat-phenomena  the  unit  mass-pair  must  be  considered  as  two 
somethings  called  molecules,  for  chemical  energy  two  somethings 
called  atoms,  for  electrical  energy  two  somethings  called  elec- 
trons, etc.,  etc. 

If  this  general  idea  be  reduced  to  a  mathematical  basis,  it 
will  appear  that  the  extent  of  mass-pairing,  or  extensity  of 
energy,  X,  of  which  any  aggregate  mass  M  is  capable,  grows 
with  the  number  of  equal  parts  n  into  which  the  aggregation  is 
subdivided,  according  to  the  equation 

X  =  4M2(1-^-)  (26) 

From  this,  if  n=i,  X=o.     If  n=2,  X=JM2.     If  n=4,  X  = 

-3_M2.     If  n=ioo,  X=o.495M2.      And  as  n  grows  indefinitely 

8 

larger  the  value  of  X  approaches  more  and  more  nearly  and 
slowly  to  JM2,  which  value  it  can  never  reach. 

On  the  other  hand,  if  n  becomes  less  than  two  the  value  of 
X  becomes  a  fraction  of  JM2.  Such  would  be  the  case  when  the 
aggregate  mass  was  divided  into  two  portions,  but  not  equal  por- 
tions, the  value  of  n  becoming  one  plus  a  smaller  and  smaller 
fraction  as  the  one  portion  became  a  smaller  and  smaller  fraction 
of  the  whole.  Such  is  the  case,  for  instance,  in  the  energetics 
of  engineering,  if  the  mass  of  the  earth  be  considered  as  the 
unit  of  measurement.  As  wre  split  off  from  the  earth  portions 
which  we  manufacture  into  cannon-balls,  railroad-trains,  etc., 
we  are  in  reality  giving  to  n  of  Equation  26  the  value  of  one 
plus  a  very  small  fraction.  The  energy  becomes  zero,  because 
its  mass-factor  has  become  zero,  only  when  the  value  of  n 
becomes  unity,  signifying  that  the  earth  is  a  homogeneous  solid, 
possessing  no  dissevered  parts  like  cannon-balls. 

If  n  becomes  less  than  unity,  the  value  of  X  becomes  nega- 
tive. Such  would  be  the  case  when  the  entire  mass  in  question 
is  but  a  part  of  a  single  member  of  some  much  larger  mass-pair. 
As  the  value  of  n,  which  must  always  be  a  positive  quantity, 
approaches  zero  the  value  of  X  approaches  negative  infinity. 

The  application  of  this  formula  to  the  understanding  of 
energetic  action  will  be  made  in  later  papers. 


58  ENERGY 

Hitherto  the  principles  of  mechanics  have  usually  been  dis- 
cussed in  the  text-books  only  in  terms  of  constant  mass-quan- 
tities. Indeed,  there  is  no  college-treatise  yet  come  to  the  writer's 
attention  which  gives  to  the  student  even  an  inkling  of  the  fact 
that  the  mass-factor  in  energy  may  be — to  say  nothing  of  the 
fact  that  it  always  is — just  as  variable  and  active  a  factor  as  the 
space  or  motion  factor.  For  the  purposes  of  applied  mechanics 
this  is  quite  sufficient;  but  so  soon  as  the  general  field  of  ener- 
getics is  entered — as  it  must  be  by  every  modern  student  in 
even  the  specialized  branches  of  engineering,  wherein  are  con- 
stantly met  transformations  between  heat,  work,  chemical  and 
electrical  energies — the  method  promptly  becomes  disqualified. 
The  student  should  be  taught,  as  soon  as  any  general  energetic 
concepts  whatever  are  presented,  which  is  usually  in  the  study  of 
thermodynamics,  in  the  junior  year,  that  the  mass-factor  of 
energy  is  just  as  frequently  and  widely  a  variable  as  is  any 
other;  and  that  the  mass-factor  is  not  merely  mass,  but  may  be 
anything  between  the  first  and  second  powers  of  mass. 

To  accomplish  this,  between  the  course  in  the  elements  of 
applied  mechanics  (usually  taught  as  a  part  of  the  general  course 
in  physics)  and  the  course  in  thermodynamics,  should  be  in- 
serted a  separate  course  upon  the  true  elements  of  mechanics  and 
mechanical  energetics,  as  outlined  above.  The  approximate  for- 
mulae of  applied  mechanics  the  student  is  to  continue  to  use  in 
his  engineering  problems.  The  true  formulae  are  to  furnish 
him  with  his  concepts  of  truly  natural  mechanical  action,  with 
which  he  is  to  interpret  the  obscure  phenomena  of  thermal, 
chemical  and  electrical  interactions. 

In  this,  the  basic  fact  now  needing  especial  emphasis  is  that 
when  energy  is  imparted  to  mass  it  may  find  embodiment  therein 
in  either  of  two  general  ways ;  and  in  nature  these  two  ways 
occur,  not  only  with  equal  frequency,  but  always  in  combination. 
First,  it  may  increase  the  relative  velocity  of  motion,  or  space 
of  separation,  of  mass-portions  which  are  already  in  separate 
existence;  that  is,  it  may  increase  the  intensity  of  energy.  Sec- 
ondly, it  may  subdivide  mass  which  was  previously  unified  into 
newly  separated  mass-pairs,  into  each  of  which  is  injected  a  first 
measure  of  relative  separation  or  relative  motion,  or  both;  that 
is,  it  may  increase  the  extensity  of  energy. 

It  has  been  too  commonly  taught  that  only  when  we  raise 


THE  TWO  FACTORS  59 

weights  or  accelerate  cannon-balls  do  we  embody  energy  in  mass, 
and  too  seldom  taught  that  when  we  crush  rock  or  grind  cement 
we  do  the  same.  Only,  when  the  cement  is  ground  or  the  rock 
crushed  the  resultant  particles  do  not  embody,  in  their  disgrega- 
tion,  an  energy  which  we  can  utilize  again.  So  that  we  say  that 
it  is  "gone";  although  all  of  it  which  has  not  become  heat  lies 
there  right  before  our  eyes.  And  even  as  to  the  heat,  that 
appears,  upon  inspection,  to  be  merely  a  finer  degree  of  comminu- 
tion and  disgregation  than  that  embodied  in  the  visible  particles. 
For  not  only  will  the  same  processes  which  grind  rock  into 
powder  also  grind  water  into  steam-heat,  but  the  latter  is,  almost 
equally  with  the  former,  unavailable  for  further  energetic  use. 

The  student  has  been  taught  from  the  start,  in  the  doctrine 
of  the  Conservation  of  Mass,  that  mass  can  be  neither  created 
nor  destroyed.  But  has  he  been  taught  with  equal  care  that 
there  is  no  known  limit  to  its  aggregation  or  subdivision?  Or 
that  these  processes  are  going  on  at  all  times,  in  nature?  Or 
that  energy  is  as  much  involved  in  these  processes  as  it  is  in  the 
accumulation  or  dissipation  of  motion  or  space  ? 

Nor  can  any  limit  be  placed  to  the  value  to  the  student  of  an 
exact  concept  of  this  great  natural  fact,  in  after  life.  It  may  be 
only  suggested  here  how  wide  may  be  its  useful  application.  It 
is  not  alone  in  thermal,  chemical  and  kindred  phenomena  that 
subdivision,  specialization  and  organization  of  subject-matter  into 
cooperation  are  of  prime  importance  for  the  effectiveness  of 
energy.  Before  the  student  has  reached  college  he  has  learned, 
upon  the  play-ground,  to  substitute  skill  for  bull-strength ;  he  has 
learned  that  athletic  energy  must  be  subdivided  and  organized 
into  team-work  before  games  can  be  won,  or  even  played,  which 
are  worth  while.  As  college-student  and  embryo  manufacturer 
he  learns  that  shop-organization  and  office-specialization  are 
prime  factors  of  success  in  business.  If  he  joins  the  militia  he 
learns  the  importance,  in  military  or  combative  energy,  of  the 
solidification  of  bodies  of  men  for  the  resistance  of  shock;  but 
of  their  subdivision  and  specialization  to  a  high  degree,  until 
virtually  each  individual  does  a  different  thing,  in  order  to 
embody  in  them  the  highest  degree  of  combative  effectiveness. 
Should  he  study  law  he  learns  from  the  most  eminent  jurists 
that  the  whole  business  of  the  law  is  to  define  and  determine  the 
relationships  which  are  to  prevail,  as  natural,  between  the 


60  ENERGY 

individuals  of  our  ever  more  numerous,  more  diversely  intricate 
and  more  energetic  race;  and  natural  relationships  are  just  what 
these  pages  are  to  define  and  make  familiar,  by  study  of  them 
as  they  exist  between  the  simplest  possible  individual  elements. 

It  is  further  the  writers'  prediction  that  before  to-day's 
student  has  reached  middle-age  he  will  have  learned,  from 
forcible  and  costly  experience  in  helping  to  make  his  country's 
history,  that  what  that  country  has  needed  most,  for  decades,  is 
a  similarly  accurate  concept  of  the  natural  relationships  between 
man  and  man,  in  the  day's  work.  He  will  have  learned  that 
what  it  has  long  urgently  needed,  and  soon  can  exist  no  longer 
without,  is  an  economic  organization,  throughout  its  every  eco- 
nomic activity,  similar  to  that  now  prevailing  within  each  factory. 
Within  each  factory  we  now  have  superlative  discipline;  but 
between  our  factories  prevails  superlative  anarchy  and  civil 
strife.  We  are  relying  too  much  at  present,  for  prospective  cure 
of  our  economic  troubles,  upon  greater  intensity  of  individual 
effort.  We  are  too  unconscious  of  the  possibilities  of  greater 
extensity  of  coordinated  energy.  We  scarcely  know  yet  what 
the  words  mean. 

Whether  a  student  is  to  become  an  engineer,  a  teacher,  a 
business-man,  a  lawyer,  a  preacher  or  a  statesman,  his  life-work 
should  be  founded  upon  a  clear  understanding  of  mechanical 
energetics.  In  each  of  those  fields,  if  his  education  is  to  aid  his 
life-work,  he  must  have  at  his  fingers'  ends  the  fact  that  energy 
can  be  accumulated  by  accumulating  mere  mass ;  but  that  then  it 
partakes  of  the  nature  of  solid  mass.  Solidity,  rigidity, 
inflexibility,  hardness  are  its  characteristics;  impact  and  friction 
are  its  results.  But  that  if,  with  the  accumulation  of  mass — or 
even  without  it — go  subdivision,  specialization  and  coordination 
into  increasing  extensity  of  mass-pairing  in  energetic  interaction, 
then  fluidity,  flexibility  and  efficiency  in  work-performance  be- 
come its  characteristics.  In  addition,  the  amount  of  energy 
which  can  then  be  embodied  in  any  given  mass  is  extended  indefi- 
nitely, absolutely  without  limit  foreshadowed  by  those  mathe- 
matical formulae  which  are  yet  available.  This  is  as  true  of 
masses  of  men  as  of  masses  of  matter. 


CHAPTER  V. 

THE  EXTREME  OR  CRITICAL  ENERGETIC  CONDITIONS. 

In  considering  the  mechanical  elements  of  the  original  freely 
moving  mass-pair,  early  in  this  series  of  papers,  it  was  established 
that  its  energy  vibrated,  during  its  progress  along  a  conic-section 
orbit,  on  either  side  of  a  mean  energetic  condition.  This  mean 
energetic  condition  was  identified  as  occurring  at  the  extremities 
of  the  latus  rectum  of  the  conic-section  orbit,  twice  in  each 
revolution. 

The  aspects  of  the  different  possible  forms  of  orbit  which 
were  listed  in  Chapter  II,  viewed  in  reference  to  this  mean 
energetic  condition,  may  be  stated  as  follows : 

1.  If  the  orbit  be  circular  all  points  of  the  orbit  embody  the 
mean  energetic  condition,  and  no  energy-transformation  what- 
ever occurs.    The  pair  is  perceptible  only  as  a  unit.     Its  orbital 
motion  is  evident  to  man  only  in  its  occupancy  of  space  and  its 
resistance  to  compression;  that  is,  by  its  permanence  and  in- 
destructibility. 

2.  If  the  orbit  be  elliptic  there  are  two  mean  energetic  points 
in  each  revolution,  through  which  energy-transformation  occurs 
periodically,  first  in  one  direction  and  then  in  the  other,  like  the 
swing  of  a  pendulum;  and  this  process  continues  indefinitely. 

3.  If  the  orbit  be  parabolic,  while  there  are  still  two  mean 
energetic  points  per  revolution,  yet  there  can  be  only  one  swing 
of  the  pendulum ;  and  on  its  outward   swing  the  energy-trans- 
formation tends  toward  a  complete  balance,   with   no   residue. 
That  is  to   say,  as  the  bodies   swing  apart,  converting  kinetic 
energy  into  potential,  the  former  is  just  almost,  but  not  quite, 
absorbed  in  accomplishing  the  utmost  separation  which  is  imag- 
inable for  the  pair.     As  the  bodies  separate  very  widely  their 
velocity  almost  becomes  zero ;  but  it  never  quite  does  so.  for  the 
gravitational  attraction  will  then  also  have  become  almost  zero, 
so  that  there  is  little  tendencv  to  stop  separating  and  reverse  into 
a   mutual   approach.     But  this   perfect  balance   of   kinetic  and 
potential  energies  may  be  regarded  as  so  very  unlikely  as  to  be 

61 


62  ENERGY 

non-existent  in  nature.    Virtually,  all  orbits  are  either  elliptic  or 
hyperbolic  in  form  and  nature. 

4.  If  the  orbit  be  hyperbolic  there  is  not  merely  only  one 
swing  of  the  pendulum  and  only  two  mean  energetic  positions, 
but  there  occurs  permanent  dissociation  of  the  pair  afterwards. 
The  kinetic  energy  at  periastron  is  so  great  that,  even  after 
separation  to  an  infinite   distance,  there  still   remains   a  finite 
residue  of  velocity  of  separation,  which  can  never  be  absorbed 
potentially  by  the  pair. 

5.  If  the  orbit  be  a  straight  line — which  is  the  limiting  case 
of  an  hyperbola  with  infinite  eccentricity — there  is  no  discernible 
mean  energetic  condition,  no  appreciable  perturbation  of  path 
by  the  pair's  propinquity  (which  is  zero),  and  no  exact  measure 
for  the  energy  involved.     All  quantities  have  passed  to  either 
zeros  or  infinities,  neither  of  which  the  mind  of  man  can  grasp. 
The  situation  is  to  be  mentioned,  not  as  a  natural  fact,  but  as  a 
mathematical   limiting  condition,   to   which   natural   phenomena 
may  never  attain;   and  also   to   remind   the   student   how   un- 
natural is  the  straight-line  path  of  motion,  followed  by  matter 
only  when  under  constant  constraint. 

Let  this  question  be  illustrated  by  means  of  a  somewhat 
familiar  mundane  mechanism. 

Imagine  a  cannon  placed  upon  a  platform  elevated  ten  miles 
above  the  surface  of  the  earth,  aimed  horizontally,  and  fired.  Its 
projectile  will  start  upon  an  orbit  which  is  commonly  described 
as  a  vertical  inverted  parabola.  But  this  is  true  only  upon  the 
erroneous  assumption  that  the  earth  is  a  flat  solid  of  indefinite 
horizontal  extent,  its  lines  of  gravitational  attraction  being  par- 
allel vertical  ones,  of  equal  intensity  at  all  distances  from  the 
earth.  But  Newton  himself  proved  that  the  gravitational  effect 
of  a  sphere  of  appreciable  dimensions  followed  his  law  exactly, 
even  when  distances  very  small  in  proportion  to  its  diameter 
were  considered,  just  as  if  the  mass  were  concentrated  at  the 
sphere's  center.  The  true  definition  of  the  projectile's  path  is 
therefore  an  ellipse,  having  its  further  focus  coincident  with  the 
earth's  center.  This  path,  however,  the  projectile  cannot  prose- 
cute far  before  it  is  interrupted  by  collision  with  the  earth's 
solid  surface. 

Suppose,  however,  that  muzzle-velocities  might  be  increased 
indefinitely.  The  process  would  in  reality  be  one  of  adding 


CRITICAL   ENERGETIC  CONDITIONS  63 

energy  to  the  pair  tangentially,  at  apastron.  The  ellipse  would 
then  grow  wider,  the  eccentricity  less,  and  the  mean  energetic 
distance  from  the  earth's  center  greater.  When  the  muzzle- 
velocity  had  attained  to  some  26,000  feet  per  second  the  elliptic 
orbit  would  have  become  widened  into  a  circle,  clearing  the 
earth's  surface  and  its  mountain-tops,  and  returning  the  pro- 
jectile to  the  point  of  its  start  once  each  eighty-five  minutes. 
If  the  cannon  had  been  wheeled  out  of  the  way  during  the 
progress  of  the  first  circuit  of  the  earth,  and  if  the  atmosphere 
were  absent,  the  projectile  would  continue  thus  as  a  satellite  of 
the  earth,  in  circular  orbit,  indefinitely.* 

Suppose  now  that  the  muzzle-velocity  of  the  projectile  might 
be  still  further  increased.  Its  orbit  will  now  have  become  elliptic 
again.  But  now  the  point  of  start  is  located,  not  at  apastron,  but 
at  periastron  of  the  orbit.  The  projectile,  at  apastron,  is  "over" 
the  antipodes  ;  and  at  each  increase  of  muzzle-velocity  its  height 
above  the  antipodes  increases.  The  eccentricity  of  orbit  now 
increases  again.  The  major  axis  of  the  elliptical  orbit  elongates, 
and  the  period  of  time  elapsing  between  each  two  returns  of  the 
projectile  to  the  platform  increases,  according  to  Kepler's  Third 
Law,  from  the  original  eighty-five  minutes  by  the  three-halves 
power  of  the  major  axis. 

Finally,  when  the  muzzle-velocity  should  have  reached  and 
surpassed  some  37,000  feet  per  second,  the  orbit  would  have 
become,  first  parabolic,  and  then  hyperbolic;  and  the  projectile 
would  depart  from  the  earth  forever.  Any  higher  muzzle- 

*This  velocity  is  derived  from  Equation  17.  Taking  the  values  for 
g  as  32.18,  for  the  radius  of  the  orbit  as  3,960  miles  and  the  mass  of  the 
earth  as  something  less  than  42Y.IO22,  it  appears  from  Equations  7  and  17 
that  the  velocities  of  equilibrium  are  as  follows:  — 

Ft.  per  sec. 

Circular    motion    around    the    earth,    just    above    its    surface,      26,000 

Vertical  motion,  directly  "up"  or  away  from  the  earth,  suf- 
ficient to  carry  the  projectile  indefinitely  into  space  —  or  contrari- 
wise, the  velocity  of  striking  the  earth  after  free  fall  from  space 
directly  toward  the  earth's  center,  ............................  37,ooo 

The  last  figure  gives  the  maximum  velocity  of  relative  motion  which 
can  exist  between  a  mass-portion  of  the  density  and  dimensions  of  our 
earth  and  a  mass-portion  considerably  smaller,  as  the  exclusive  property 
of  the  pair,  and  which  can  continue  indefinitely,  without  dissipation  in 
either  collision  or  dissociation.  Should  hisrher  velocities  than  this  be 
observed  they  must  be  the  property  not  of  the  mass-oair  between  which 
they  are  observed  by  the  eye,  but  of  some  much  larger  mass-system. 
Only  such  a  larger  mass-system  could  either  generate  or  arrest  such 
excessive  velocities. 


OF  "HE 

UNIVERSITY 

OF 


64  ENERGY 

velocities  than  this  would,  of  course,  only  accentuate  the  prompt- 
ness of  this  dissociation. 

For  any  given  mass-pair — and  virtually,  for  any  given  major 
mass,  when  the  individual  masses  are  widely  dissimilar — this 
"critical"  velocity  at  periastron,  above  which  dissociation  takes 
place,  is  a  definite  thing,  for  any  given  periastron  distance  S0. 
For,  as  was  noted  in  the  preceding  paper,  the  expression  for  the 

spacial  intensity  of  energy  of  a  pair  is  c  (g -g  )  ;  and  if,  in  this, 

S  becomes  very  great  indeed,  the  intensity  takes  the  definite 
value  -~-.  Therefore,  if  two  bodies  fall  together  from  any  dis- 
tance of  separation  whatever,  however  great,  into  a  minimum 
separation  of  S0,  this  is  the  utmost  intensity  of  motion  which 
they  can  develop.  In  the  case  of  the  earth  this  velocity  is  about 
37,000  feet  per  second.  Conversely,  two  bodies  situated  at  the 
minimum  distance  of  S0  require  only  this  intensity  of  motion  to 
dissociate  them  permanently. 

This  degree  of  intensity,  then,  is  the  utmost  which  this  par- 
ticular pair  is  capable  of  embodying,  unless  conditions  permit 
them  to  approach  nearer  than  the  distance  S0.  Ordinarily,  S0  is 
limited  by  the  finite  dimensions  of  the  bodies,  when  solid ;  but 
theoretically  there  is  no  limit  to  the  diminution  of  S0,  and  as  it 
diminishes  the  intensity  of  energy  increases  very  rapidly. 

But  at  present  it  is  not  so  important  to  discuss  the  possible 
variations  of  S0  as  it  is  to  note  that,  if  any  pair  be  observed  at 
periastron  with  an  intensity  of  motion  greater  than  the  "critical'' 

Q 

one,  corresponding  to-~-,  implying  a  hyperbolic  orbit  and  prompt 
dissociation,  it  signifies  that  the  pair  visualizes  what  it  cannot, 

Q 

of  itself,  embody,  viz :    a  fund  of   intensity  greater  than  ^-* 

The  surplus  of  energy  over  this  quantity  cannot  possibly  be  the 
property  of  the  pair  itself.  It  is  a  manifestation  by  the  pair  of 
an  energy-fund  held  by  it  in  conjunction  with  some  third  ex- 
ternal mass-portion,  which  third  portion  is  itself  not  necessarily 
visible  directly  as  a  member  of  the  system.  But  nothing  but 
some  such  larger  third  mass-portion  is  capable  of  destroying  the 


CRITICAL  ENERGETIC   CONDITIONS  "65 

surplus  velocity,  and  none  but  it  could  have  been  capable  of 
producing  it. 

This  surplus,  or  third-mass,  energy  may  be  called,  for  the 
present,  the  "external"  energy  of  the  pair,  in  order  to  keep  our 
argument  in  terms  of  a  simple  mass-pair  as  a  base.  This  ex- 
ternal energy  the  pair  has  'borrowed  and  displayed  as  its  own, 
so  to  speak,  although  it  has  no  rightful  ownership  of  it  and  must 
soon  repay  the  loan. 

The  relation  of  such  external  obligations  to  its  own  internal 
assets  of  energy  is  visible  directly  in  the  value  of  the  eccentricity 
of  orbit  e.  For  it  is  only  when  e  is  greater  than  unity  that 
hyperbolic  motion  and  dissociation  can  occur.  The  excess  of  e 
above  unity,  therefore,  is  a  measure  of  the  external  or  borrowed 
energy  which  is  displayed  by  the  pair. 

In  order  to  understand  this  situation  completely,  return  must 
be  made  for  a  moment  to  the  question  of  radial  and  tangential 
energies. 

A  value  of  e  greater  than  unity  implies  an  angle  between 
mean  energetic  motion  and  the  radius  vector  connecting  the  two 
bodies  (see  Fig.  4)  smaller  than  45°.  In  that  case  the  radial 
component  of  motion  would  be  greater  than  the  tangential 
Here,  again,  one  must  be  careful  with  his  mathematical  ex- 
pressions for  energy.  If  it  be  assumed,  too  readily,  that  kinetic 
energies  are  absolutely  proportional  to  the  square  of  the  velocity, 
then  the  ratio  of  radial  to  tangential  kinetic  energies,  at  the  mean 
energetic  point,  must  be  £2;  for  c  =  cotan  a,  the  ratio  of  radial 
to  tangential  velocity  at  this  point. 

But  two  objections  exist  to  this  reasoning.  In  the  first  place, 
no  exact  expression  for  the  tangential  energy  exists  at  all.  In 
the  second  place,  if  there  were  one  it  must  be  negatively,  rather 
than  directly,  proportional  to  the  velocity  squared;  for  energy 
must  be  abstracted  in  order  to  increase  the  tangential  velocity. 

The  way  out  of  this  puzzle  is  to  note,  from  Equation  22, 

Radial  Energy  =  2  c  MtM2  -^  =  2  e  MtM2  -?'  sin'  *        (22) 

- 


that,  for  any  stated  mean  energetic  conditions,  such  as  distance 
D  and  tangential  velocity  U  sin  a,  the  radial  energy  is  directly 
proportional  to  the  eccentricity  of  orbit  e.  And  since  the  mass- 
pairing  factor  M1M2  is  always  alike  on  both  sides  of  this  equa- 


66  ENERGY 

tion,  bearing  no  influence  upon  the  result,  it  would  have  been 
more  explicit  if  this  statement  had  been  made  in  terms  of 
intensities  of  energy,  rather  than  of  energy  itself.  Thus  it  may 
be  considered  proven  that  when  the  radial  intensity  —  which  has 
already  been  characterized  as  the  'medium  of  communication,  or 
manifestation,  between  the  pair  and  the  external  world  of  mass  — 
assumes  a  proportion  to  the  internal,  or  stay-at-home,  intensity 
greater  than  equality,  then  dissociation  and  a  cessation  of  the 
pair's  visible  existence  must  ensue  promptly.  It  is  with  a  mole- 
cule much  as  it  is  with  a  man  :  when  he  goes  too  energetically  into 
foreign  politics,  to  the  overshadowing  of  his  private  business,  his 
home  breaks  up. 

The  Lower  Critical  Intensity.  It  has  already  been  sug- 
gested, however,  that  something  else  than  dissociation  may 
happen  to  interrupt  the  continuity  of  the  pair's  energetic  existence 
in  smooth,  unbroken  orbit,  with  energy  manifestly  conserved. 
Collision  may  occur,  at  or  just  before  periastron. 

For  this  to  take  place,  the  distance  of  separation  at  periastron 
must  be  less  than  the  sum  of  the  radii  of  the  two  bodies.  The 
situation  must  therefore  be  investigated  through  the  energetic 
equation  in  terms  of  periastron  conditions,  or  Equation  23. 


The  last  two  terms  of  this  equation  readily  reduce  to 

c  V2  1 

——=-——      _i_  =  T 

S    "" 


.   (23) 

(27} 

which  is  the  fundamental  equation  for  the  critical  intensity,  in 
which  the  first  two  terms  define  the  limits  of  spacial  concentra- 
tion, on  the  one  hand,  or  of  embodiment  of  internal  motion,  on 
the  other,  which  respectively  constitute  the  criteria  of  energy- 
transformation. 

If,  in  Equation  27,  S0  be  considered  equal  to  the  sum  of  the 
radii  of  the  two  bodies,  the  expression  gives  the  limiting  condi- 
tion by  which  collision  at  periastron  may  be  avoided. 

The  Critical  Limits  of  Intensity.  Upon  these  considera- 
tions may  be  founded  the  statement  that  the  determining  factor 
in  the  permanence  of  energy  is  its  intensity.  Within  certain 
limits  the  intensity  of  a  given  pair  may  vary  ;  with  effect  upon  the 
external  appearance  of  the  energy,  it  is  true,  but  without  pre- 


CRITICAL   ENERGETIC   CONDITIONS  67 

cipitating  a  transformation  of  the  energy  into  some  other  appar- 
ently independent  form.  But  as  soon  as  these  limits  are  sur- 
passed, the  energy  alters  its  outward  aspect  so  radically  as  to 
make  it  difficult  to  discern  the  true  continuity  of  its  existence. 
We  are  constrained  to  say  that  the  energy  has  therein  become 
"transformed,"  so  that  we  give  it  another  name  than  before. 
And  as  the  understanding  of  the  more  obscure  forms  of  energy 
is  inextricably  connected  with  energy-transformation,  which  gives 
them  birth  and  death,  it  is  clear  that  the  entire  science  of  ener- 
getics turns  upon  these  critical  limits  of  intensity  as  a  car  does 
upon  two  king-pins. 

These  limits,  and  their  effects,  may  be  stated  briefly  as 
follows : 

1.  If  the  kinetic,  or  outward  radial,  intensity  exceed  the 
limit  implied  by  the  condition  e=i,  the  pair  will  dissociate. 

2.  If  the  spacial,  or   inward   radial,  intensity    exceed  the 

limit  implied  by  the  condition    — ==  wherein  the  R's  are 

k0     K-nr, 

the  radii  of  the  two  masses,  the  pair  will  collide.  (For  the 
intensity  increases,  it  will  be  remembered,  as  S0  grows  smaller.) 

Contrasting  these  two  sorts  of  critical  condition,  in  terms  of 
the  contrasted  aspects  of  the  two  extremes  of  the  simple  orbit, 
the  first  is  perceptible  as  a  critical  limit  to  the  spacial  expansion 
of  the  system,  by  too  great  velocity  radially  outward — in  which 
phenomenon  force  plays  little  part.  The  second  is  perceptible  as 
a  critical  limit  to  the  forceful  compression  of  the  system,  by  too 
great  a  velocity  radially  inward — in  which  phenomenon  space 
plays  little  part.  It  will  develop  later  that  the  first  form  of 
critical  condition  is  of  interest  in  connection  with  the  too  great 
expansion  of  vapors  and  gases.  The  second  is  of  interest  in 
connection  with  the  too  forceful  compression  of  solids.  For 
either  procedure  will  develop  energy-transformation  in  bodies  of 
matter. 

After  dissociation  occurs,  the  further  history  of  the  pair  is 
about  as  easily  written  as  is  that  of  the  snakes  in  Ireland.  Vir- 
tually speaking,  there  is  no  further  history.  The  pair  has  ceased 
to  exist,  as  a  pair,  though  its  two  members  continue  an  inde- 
structible existence. 

Yet  in  this  facile  statement,  which  is  necessary  in  order  to 
take  the  road  step  by  step,  lies  in  reality  a  great  error  against 


68  ENERGY 

which  the  student  should  be  warned.  In  truth,  however  dis- 
tantly (and  apparently  permanently)  a  mass-pair  may  be  sepa- 
rated, by  chance  foreign  influence,  yet  it  is  never  completely 
sundered,  and  never  can  be.  Though  the  distance  of  separation 
may  become  that  which  now  lies  between  any  object  at  hand 
and  those  scattered  throughout  the  farther  heavens,  at  distances 
beyond  human  comprehension,  yet  the  mutual  bond  of  attraction 
still  exists,  in  the  case  of  each  pair,  to  a  measurable  degree. 
And  it  exists  eternally.  Though  the  number  of  centuries  which 
must  elapse,  before  chance  again  frees  the  pair  for  a  fall  into 
mutual  propinquity  and  perceptible  energetic  reaction,  may  far 
surpass  human  understanding,  and  strain  even  our  arithmetic  to 
compass  its  expression,  yet  when  the  time  does  come  the  latent 
mutual  affection  will  be  just  as  potent  for  warmth  of  greeting 
as  if  it  occurred  to-day. 

As  to  the  further  results  of  collision,  on  the  other  hand,  that 
topic  lies  much  nearer  to  the  purpose  of  these  papers  than  does 
dissociation.  Indeed,  its  bearing  is  so  vital  that  it  will  be  post- 
poned for  a  special  chapter  of  its  own.  In  the  meantime,  atten- 
tion may  be  turned  to  those  processes  which  may  lead  the  energy 
of  any  mass-pair  to  exceed  the  critical  limits  of  kinetic  or  spacial 
intensity,  with  the  results  just  defined. 

Collision  and  consolidation  of  the  pair  into  unity  may  result 
from  the  abstraction  of  energy  in  either  of  two  ways,  viz : 

1.  .If  the  energy  be  extracted  in  radial  form  (that  is,  between 
periastron  and  apastron,  by  a  medium  capable  of  absorbing  only 
the  radial  component  of  motion),  the  eccentricity  e  decreases. 
Because  the  tangential  component  has  remained  unaffected,  the 
mean  energetic  distance  D  will  remain  unaltered.     The  motion 
becomes  more  nearly  circular,  and  thus  lapses  toward  a  peaceful 
consolidation ;  for  it  was  explained  in  an  earlier  paper  that  true 
circularity  of  motion  constituted  an  apparent  unity. 

2.  If  the  energy  be  abstracted  in  tangential  form   (that  is> 
at  apastron,  by  a  medium  capable  of  absorbing  only  tangential 
energy),  the  eccentricity  e  is  increased   while  the  mean  energetic 
distance  D  is  decreased.     From  both  reasons  the  periastron. dis- 
tance S0  decreases.    The  motion  becomes  more  nearly  rectilinear, 
and  may  impinge  upon  the  solid  confines  of  the  bodies.     Thus 
would  ensue  a  violent  consolidation  of  the  pair. 

Thus,   in   the   illustration    of    the   cannon-ball,    velocities   at 


CRITICAL  ENERGETIC  CONDITIONS  69 

periastron  varying  only  between  about  26,000  and  37,000  feet 
per  second  would  permit  permanency  of  orbit.  Anything  higher 
would  lead  to  dissociation ;  anything  lower  to  collision.  If,  when 
velocities  were  between  these  critical  conditions,  the  radial  com- 
ponent only  of  motion  should  be  abstracted,  the  orbit  would 
reduce  toward  a  circle  of  large  radius  about  the  earth,  at  a  rate 
of  26,000  feet  or  less  per  second.  The  speed  mentioned  would 
then  be  the  one  of  equilibrium,  and  the  pair  would  exist  per- 
manently, as  a  "unit,"  in  this  condition,  until  again  disturbed 
from  without. 

But  this  process  of  abstracting  energy  radially  could  never 
be  carried  to  the  point  where  the  orbit  became  truly  circular, 
with  zero  eccentricity.  For  energy  might  be  abstracted  radially 
only  to  the  extent  that  a  radial  component  of  motion  were 
present.  As  the  orbit  approached  circularity  radial  energy  could 
be  abstracted  only  with  greater  and  greater  difficulty,  or  under 
more  and  more  unusual  conditions.  The  process  could  not  imag- 
inably proceed  until  all  eccentricity  were  gone. 

If,  on  the  other  hand,  the  energy  were  abstracted  tangentially 
at  apastron,  or  at  any  rate  during  the  outer  half  of  the  elliptic 
orbit,  the  latter  would  become  a  more  narrow  ellipse,  like  that 
of  any  cannon-ball,  and  collision  with  the  earth  would  ensue 
quite  as  in  any  such  a  case. 

These  actions  can  perhaps  be  better  understood  from  Fig.  5, 
which  displays  the  several  possible  orbits  of  a  body  Mj  relatively 
to  a  larger  mate.  Around  the  mate  is  shown  a  dotted  circle  ZZ, 
the  radius  of  which  is  the  sum  of  the  radii  of  the  (supposedly 
spherical)  bodies.  Any  orbit  which  touches  this  circle  will  of 
course  end  in  collision.  Thus,  AA  and  FF  are  both  hyperbolic 
orbits ;  but  one  of  them  will  end  in  dissociation,  the  other  in 
collision. 

Should  radial  energy  be  abstracted  from  AA  by  radial  means, 
between  P  and  A,  the  orbit  would  be  altered  to  some  hyperbola 
or  ellipse  of  less  eccentricity  than  AA  and  passing  through  the 
point  Mj ;  that  is,  the  mean  energetic  distance  would  be  con- 
served. The  limiting  case  for  such  a  process,  when  all  the 
radial  energy  had  been  withdrawn,  would  be  the  circle,  which 
is  the  only  one  of  these  orbits  of  lesser  eccentricity  which  is 
shown. 

The  withdrawal  of  tangential  energy  from  A  A  would  lead  to 


70 


ENERGY 


a  hyperbola  of  increased  eccentricity,  passing  through  M^  and 
cutting  the  circle  ZZ. 

At  periastron  P  the  radial  energy  of  A  A  (as  was  pointed  out 
in  an  earlier  paper)  is  all  kinetic  in  form;  and  that  kinetic 
energy  is  directed  tangentially.  Yet  all  of  it  above  the  velocity 
for  circular  equilibrium  at  this  radius  is  true  radial  energy;  for 


its  radial  or  centriiugal  force  is  unbalanced,  and  the  motion- 
energy  will  quickly  become  radial  in  direction  as  well  as  name. 
Therefore,  if  kinetic  energy  be  abstracted  at  P,  it  amounts  to  a 
reduction  of  radial  energy.  Radial  energy  will  have  been 
reduced,  though  not  by  radial  action.  The  eccentricity  will 
decrease  and  the  orbit  finally  become  an  ellipse,  such  as  BB. 
Further  abstractions  of  kinetic  energy  at  P  will  alter  the  orbit 
to  a  circle,  such  as  CC,  or  an  ellipse  of  reversed  eccentricity, 
such  as  EE,  which  ends  in  collision. 

The  abstraction  of  kinetic  energy  at  the  apastron  Q  of  BB, 


CRITICAL  ENERGETIC  CONDITIONS  71 

on  the  other  hand,  increases  the  eccentricity,  but  decreases  the 
periastron  distance,  as  in  the  orbit  GG,  and  leads  to  collision. 
For  at  Q  the  kinetic  energy  is  all  tangential  energy,  the  radial 
energy  being  space-energy. 

Now  in  the  illustration  of  the  cannon-ball  the  energy  was  all 
supposed  to  be  supplied  at  periastron,  as  at  P  for  the  orbits  CC, 
BB  or  AA.  But  in  the  use  for  which  these  arguments  are 
designed,  as  mechanical  similes  for  molecular  action,  all  contribu- 
tions or  abstractions  of  energy  will  be  by  other  systems  which  are 
external  to  the  one  in  question.  The  point  of  contact  between 
the  two  systems  will  be  at  or  near  apastron,  such  as  Q  for  BB  or 
GG,  or  P  for  EE.  The  only  way  in  which  tangential  energy- 
can  be  exchanged  is  tangentially  and  kinetically;  and  even  then 
it  can  be  exchanged  only  through  the  medium  of  conversion  into 
or  from  radial  energy. 

Radial  energy,  on  the  other  hand,  cannot  only  be  exchanged 
directly  with  external  systems,  but  it  can  be  exchanged  in  two 
distinct  and  different  ways.  These  ways  are  (i)  spatially,  by 
radial  action  between  periastron  and  apastron,  and  (2)  kinet- 
ically, by  tangential  action  at  periastron  or  apastron.  Referring 
to  the  illustration  of  the  cannon-ball,  the  first  would  be  instanced 
by  the  interaction  of  the  projectile,  during  its  outward  or  inward 
flight,  with  some  mass  which  was  itself  moving  outwardly  or 
inwardly,  relatively  to  the  earth.  The  second  would  be  in- 
stanced by  the  original  impulse  of  the  projectile,  by  the  gun- 
powder, or  by  some  similar,  but  negative,  influence  experienced 
from  some  third  mass  at  the  extremity  of  its  flight  away  from 
the  earth. 

This  question  of  the  possible  ways  by  which  energy  may  be 
gained  or  lost  by  a  mass-pair  should  be  clearly  understood. 
They  may  be  stated  briefly  as  follows,  although  their  significance 
will  not  come  out  until  the  discussion  reaches  the  question  of  the 
mechanical  theory  of  heat.  Thus,  energy  may  be  gained  or 
rejected  by  a  mass-pair  in  three  different  ways,  each  of  which 
will  have  a  different  effect  upon  the  form  of  orbit  and  upon  the 
chances  of  its  alteration  past  one  of  the  critical  points  of 
intensity. 

i.  Energy  received  tangentially  at  apastron  (as  in  the  illus- 
trative cannon-ball  before  its  trajectory  had  cleared  the  earth) 
increases  the  mean  energetic  distance  D  and  decreases  the  eccen- 


72  ENERGY 

tricity  e.  Energy  imparted  tangentially  at  apastron,  of  course, 
reverses  this  rule. 

2.  Energy   received   tangentially   at   periastron    (as   in   the 
illustrative  cannon-ball  after  its  trajectory  had  cleared  the  earth) 
increases  both  the  mean  energetic  distance  and  the  eccentricity. 
Energy  lost  in  similar  fashion  reverses  this  rule. 

3.  Energy  received  radially,  between  one  extreme  energetic 
condition  and  the  other,  increases  the  eccentricity,  but  does  not 
affect  the  mean  energetic  distance.     Energy  expended  radially 
decreases  the  eccentricity  without  affecting  the  mean  energetic 
distance. 

In  the  illustration  of  the  cannon-ball,  by  which  the  natural 
action  of  a  free  projectile  was  sought  to  be  made  clear,  there 
was  used,  it  is  true,  an  original  impulse,  viz:  the  energy  of  the 
gunpowder,  which  is  foreign  to  the  simple  interaction  of  two 
mass-portions  by  gravitation  and  inertia.  Yet  the  illustration 
may  not,  for  this  reason,  be  outlawed ;  for  in  the  interaction  of  a 
projectile  with  another  mass-system  consisting  of  more  than  one 
mass-portion  there  may  arise  a  form  of  impulse  which  closely 
resembles  that  of  gunpowder  and  cannon. 

If  we  revert  to  the  simple  energetic  system  displayed  in  Fig. 
2.  and  imagine  M2  to  be,  instead  of  a  doughnut-like  solid,  a  ring 
of  separate  solid  mass-portions,  revolving  about  the  center  C  in  a 
swarm,  in  circular  orbits,  then  the  situation  becomes  more  com- 
plex, and  capable  of  activities  beyond  those  already  described. 
Let  it  further  be  imagined  that  Mlt  instead  of  following  the 
straight  line  ACB  back  and  forth,  follows  some  conic-section 
orbit  which  carries  it  periodically  through  the  point  C. 

Now  the  intensity  of  action  at  this  point  was  found  to  be 

proportional   to  -=-,   wherein    S0   is    the   minimum   distance   of 

^o 

approach  between  M±  and  M2.  But  when  M2  is  an  annular  body 
such  as  shown  in  Fig.  2,  the  minimum  value  for  S0  is  the  radius 
of  the  annulus.  Therefore,  if  the  annulus  should  contract  dur- 
ing any  of  M/s  circuits  of  its  orbit,  the  intensity  of  action  at  its 
next  approach  to  M2  would  be  accentuated.  The  energy  released 
from  Mo  by  the  mutual  approach  of  its  members  would  be 
transferred  to  Mlt  imparting  to  it  a  greater  velocity;  and  this 
increase,  occurring  at  periastron,  must  be  radial  in  its  character. 


CRITICAL   ENERGETIC   CONDITIONS  73 

It  will  go  to  increase  both  the  mean  energetic  distance  and  the 
eccentricity  of  orbit  between  Mt  and  M2. 

If  the  argument  be  transferred  to  Fig.  3  new  possibilities 
appear.  If  M2  thereof  should  consist  of  two  or  more  portions, 
held  apart  by  their  relative  motion  in  elliptic  orbit,  the  degree  of 
propinquity  at  periastron,  or  the  intensity  of  energy  of  the 
MjMjj-system,  would  depend  upon  whether  the  two  or  more 
orbits  all  reached  periastron  simultaneously  or  not.  Should  M2 
be  in  its  condition  of  maximum  separation  of  parts  when  Mx 
approached,  the  intensity  of  action  would  be  slight.  But  if  all 
orbits  coincided  in  phase,  reaching  greatest  propinquity  simulta- 
neously, the  intensity  would  be  very  much  greater.  The  impulse 
received  by  Mx  in  any  such  a  way  would  be  quite  the  parallel 
of  the  impulse  received  by  the  illustrative  cannon-ball  from  the 
gunpowder,  at  periastron. 

Thus,  from  period  to  period  of  the  MjMjj-system  the 
intensity  of  its  radial  energy  might  vary  widely.  If  M±  were  the 
messenger  carrying  a  manifest  of  the  system's  energy  to  foreign 
parts,  which  could  not  directly  perceive  the  more  massive,  tan- 
gential and  torpid  energy  of  M2,  the  system  would  be  observed 
thereby  as  embodying  widely  varying  intensities  of  energy. 
While  at  one  time  the  intensity  might  be  so  limited  as  to  confine 
Mx  to  an  elliptic  orbit,  at  another  it  might  project  it  with  an 
hyperbolic  orbit.  Or,  conversely,  the  projectile  M1?  having  entered 
the  system  from  foreign  parts  on  an  hyperbolic  orbit,  might 
be  entrapped  and  remain  in  elliptic  motion.  Indeed,  its  own 
arrival  might  be  the  cause  of  such  dissociation  between  the  parts 
of  M2,  at  the  expense  of  M/s  energy,  that  the  latter  no  longer 
possessed  sufficient  energy  to  get  away  again. 

Of  the  portions  of  the  orbit  further  from  periastron,  those 
beyond  the  mean  energetic  condition  are  most  illustrative ;  though 
ail  that  is  said  also  applies  to  points  inside  the  mean. 

In  its  outer  portions  the  orbit  may  be  perturbed  by  the 
approach  of  some  third  mass-portion  arriving  as  a  messenger 
from  more  distant  fields.  This  third  projectile  wiH  possess  a 
motion  aimed  more  or  less  directly  at  the  mass-center  of  the 
MjMg-system.  To  the  extent  of  its  component  thus  directed  it 
will  exchange  radial  energy  with  the  system.  To  the  extent  of 
its  component  normal  to  this  direction  it  will  exchange  tan- 


74  ENERGY 

gential  energy.     The  effects  of  these  exchanges  upon  the  form 
of  the  original  orbit  have  already  been  noted. 

It  is  not  possible  to  trace  here  all  the  varied  possibilities 
of  such  interchanges  of  energy,  even  when  occurring  between 
systems  of  only  three  or  four  mass-portions.  What  is  desired 
is  merely  to  impart  an  elementary  concept  of  the  distinctions 
between  radial  and  tangential  energies,  and  the  ways  in  which 
these  may  be  affected  by  other  systems  without  infraction  of 
the  Conservation  of  Energy.  It  is  now  plain  that  such  outside 
influences  might  alter  widely  the  intensity  and  eccentricity  of 
any  energy-system,  leading  it  toward  or  over  either  of  the  two 
critical  limits  of  intensity  as  defined  above.  It  will  be  under- 
taken later  to  show  that  such  outside  influences  may  also  affect 
just  as  widely  the  extensity  of  a  mass-system,  though  the  process 
is  not  quite  so  simple  and  obvious. 

This  completes  the  bare  statement  of  what  mechanical  energy 
is,  and  by  what  methods  only  it  may  be  imagined  as  augmented 
or  diminished.  Whatever  hypotheses  as  to  other  forms  of 
energy  than  mechanical  may  be  made,  if  the  latter  are  to  be 
regarded  at  all  as  mechanical  energy  in  disguise  their  activities 
must  be  brought  into  line  with  the  preceding  analysis. 

For  ttfe  present  it  suffices  to  point  out  that  there  has  already 
resulted  from  this  analysis  a  useful  classification  of  mechanical 
energy-forms,  which  may  be  given  a  habitation  and  a  set  of 
names  in  the  following  table,  as  "permanent,"  "subpermanent" 
and  "superpermanent"  types  of  energy,  respectively. 

1.  The  permanent  forms  of  mechanical  energy   are  those 
embodied  in  elliptic  orbits  of  sufficient  dimensions  to  clear  the 
solid  confines  of  the  two  bodies.     The  mathematical  conditions 
defining  this  class  are  that  the  eccentricity  e  shall  be  less  than 
unity  and  the  periastron  distance  S0  shall  be  greater  than  z,  when 
z  is  the  sum  of  the  radii  of  the  two  solid  masses. 

2.  The  subpermanent  forms  of  mechanical  energy  are  those 
embodied   in   orbits,    either   elliptic   or   hyperbolic,    which   pass 
within  the  solid  confines  of  the  bodies  and  end  in  collision.    The 
mathematical  conditions  defining  this  class  are  that  e  may  have 
any  value  whatever,  but  S0  must  be  less  than  z. 

3.  The   superpermanent    forms    of   mechanical   energy    are 
those   embodied   in   hyperbolic   orbits   only,   when   of   sufficient 


CRITICAL   ENERGETIC   CONDITIONS  75 

periastron  distance  to  clear  the  solid  confines  of  the  bodies. 
They  end  in  dissociation,  instead  of  collision.  The  mathematical 
conditions  defining  this  class  are  that  e  shall  be  greater  than  unity 
and  S0  greater  than  z. 

It  will  develop  later  that,  whereas  all  the  so-called  "me- 
chanical" energies  of  the  engineer  must  belong  solely  to  the 
second  of  these  three  classes  of  energy,  the  molecular  or  atomic 
or  electronic  energies,  which  we  call  heat,  chemical  energy, 
electricity,  etc. — if  they  can  be  regarded  as  modes  of  mechanical 
motion  at  all — must  belong  to  the  first  and  third  types.  As 
to  electrical  energy  there  is  more  doubt,  because  lack  of 
permanence  is  its  chief  characteristic ;  yet  in  electrical  matters 
all  questions  of  time  must  be  referred  to  such  exceedingly  minute 
units  that  it  may  appear,  upon  examination,  that  electricity,  like 
light,  is  in  reality  one  of  the  most  permanent  of  all  forms  of 
energy. 

In  the  distinctions  portrayed  in  Fig.  5,  therefore,  we  are  on 
the  edge  of  understanding  that  most  wonderful  and  significant 
of  all  natural  phenomena,  the  transformation  of  energy.  For, 
aside  from  the  contrasts  between  the  kinetic  and  potential,  and 
the  radial  and  tangential — sorts  which  form  a  part  of  every  type 
of  energy — no  alterations  in  form  of  action  have  developed,  thus 
far  in  the  analysis,  which  are  so  subversive  of  external  appear- 
ance as  are  these  changes  from  permanent  to  sub-  or  super- 
permanent  conditions. 

The  dividing  lines  between  the  permanent,  as  the  central  form 
of  energy,  and  the  sub-  and  superpermanent  forms  on  its  either 
hand,  are  called  the  critical  energetic  conditions — the  lower  and 
upper  critical  intensities  of  energy,  respectively.  These  critical 
limits  have  already  been  briefly  defined  in  this  chapter.  The 
part  they  play  in  energetic  phenomena  cannot  be  discussed  until 
other  forms  of  energy  than  the  mechanical  are  discusssed. 

The  upper  critical  intensity  has  long  been  recognized  and 
taught — although  chiefly  of  interest  in  astrophysics — as  the 
"critical  velocity."  For  each  mass-portion,  it  has  been  taught, 
there  exists  a  certain  velocity  for  smaller  mass-portions,  above 
which  the  latter  would  dissociate  from  the  former.  If  this  idea 
be  made  more  accurate  by  defining  the  critical  function  as  a 

certain    value    of    - ,  instead    of    merely    velocity,    and 


76  ENERGY 

broadened  by  recognizing  it  as  an  intensity  of  motion-energy,  it 
can  remain  for  present  purposes  unchanged. 

The  lower  critical  intensity,  on  the  other  hand,  has  not  been 
similarly  recognized  and  taught,  so  far  as  the  ordinary  text- 
books give  evidence.  It  is  directly  proportional  to  intensity  of 

space-energy,  or  degree  of  propinquity,  or  solidarity,  __L,  accord- 
So 

ing  to  the  several  ways  in  which  it  might  be  named.  For  every 
mass-pair  of  specified  density  of  members,  therefore,  there  is  a 
certain  degree  of  intensity  of  concentration  (which  is  itself  a 
form  of  energy)  which  cannot  be  exceeded  without  entailing 
collision  and  energy-transformation. 

In  all  of  this  discussion  it  must  have  become  long  since 
obvious  that,  in  the  natural,  mechanical  interaction  of  mass- 
portions,  the  straight  line  plays  a  very  subordinate  part,  if  it 
appears  at  all.  As  a  matter  of  fact,  it  does  not  appear  at  all. 
It  has  already  been  clearly  shown  how,  mathematically  speaking, 
the  straight  line  constitutes  one  unattainable  limit  of  eccentricity 
of  orbit,  on  one  side,  while  the  circle  constitutes  another  equally 
unattainable  limit  of  eccentricity  (namely,  zero  eccentricity)  on 
the  other.  Centrally  between  the  two  lies  the  PARABOLA,  with 
unit-eccentricity,  constituting  the  geometrical  fundament  of  nat- 
ural energetics.  It  would  be  the  next  step  of  development  of 
this  question  of  the  absorption  and  rejection  of  energy  by  mass- 
systems  to  show  that  this  mathematical  aspect  of  the  situation  is 
also  the  natural  one — except  that  in  mathematics  limits  are 
attainable,  whereas  in  nature  they  are  not. 

For  it  is  a  fact,  in  the  natural  aspect  of  the  question,  that  the 
more  radial  energy  a  system  possesses,  the  more  readily  it  will 
reject  or  impart  energy  to  other  systems,  and  the  less  readily  it 
will  receive  it  in  radial  form.  Conversely,  the  more  tangential 
energy  a  system  embodies  the  more  readily  it  will  receive  and 
absorb  radial  energy,  and  the  less  readily  it  will  impart  it.  It 
follows,  therefore,  that  the  smaller  the  eccentricity  becomes,  the 
greater  difficulty  is  there  in  reducing  it  further,  and  the  greater 
are  the  chances  that  the  eccentricity  will  increase,  in  any  ener- 
getic mix-up,  rather  than  still  further  decrease.  Conversely,  the 
greater  the  eccentricity  becomes,  the  greater  is  the  likelihood  of 


CRITICAL  ENERGETIC   CONDITIONS  77 

its  being  decreased,  in  any  energetic  encounter,  rather  than  being 
still  further  increased. 

The  result  of  this  view  of  the  case  is  to  place  the  medium 
value  for  the  eccentricity — unity — in  the  most  conspicuous  posi- 
tion, as  the  value  forming  a  center  of  stable  equilibrium,  on 
either  side  of  which  the  variations  in  eccentricity  swing,  as  does 
a  pendulum  about  its  vertical  position.  Just  as  it  was  found  (see 
Chapters  II  and  III)  that  both  velocity  and  spacial  separation,  in 
energetic  systems,  swing  in  stable  equilibrium  on  either  side  of 
mean  energetic  values  for  both  variables — on  either  side  of  a 
mean  energetic  distance  of  separation  and  a  mean  energetic 
velocity,  neither  of  which  could  ever  attain  to  either  zero  or 
infinity — so  now  it  appears  that  the  eccentricity  of  natural  orbit 
also  varies  on  either  side  of  its  mean  energetic  value. 

This  mean  energetic  eccentricity  of  orbit  is  unity,  defining 
the  parabola.  This  is  the  mean  or  average  energetic  condition 
of  every  mass-pair  in  the  universe.  A  mass-pair  in  this  mean 
condition  of  eccentricity  would  be  just  upon  the  line  between 
confining  its  energies  at  home,  in  stable  permanence,  and  sending 
them  abroad.  From  this  mean  the  eccentricity  may  be  reduced 
into  elliptic  motion  by  the  abstraction  of  energy  from  the  system. 
But  such  abstraction  of  energy  becomes  more  and  more  difficult 
as  it  proceeds,  and  no  imaginable  natural  conditions  may  ever 
be  defined  which  would  succeed  in  reducing  the  eccentricity  to 
zero,  in  circular  motion. 

Similarly,  the  absorption  of  energy  by  any  mass-pair  having 
the  mean  energetic  eccentricity  of  orbit,  or  following  a  parabolic 
orbit,  may  increase  that  eccentricity  into  hyperbolic  motion 
indefinitely.  But  the  difficulty  of  further  absorption  of  energy 
increases  as  it  proceeds,  and  no  imaginable  natural  conditions 
may  be  defined  which  would  succeed  in  expanding  the  eccentricity 
to  infinity,  or  developing  straight-line  motion. 

It  is  therefore  obvious  that  the  teaching  of  mechanics  to 
mature  students  by  basing  everything  upon  straight-line  motion  is 
unnatural  to  the  last  degree.  The  fundament  of  all  true  mechanics 
is  parabolic  motion.  That  stands  to  all  other  forms  of  orbit  as 
our  sun  stands  to  all  possible  vagaries  of  its  innumerable  family 
of  satellites — as  a  natural  base  and  center  of  equilibrium  which, 
albeit  itself  unsupported  and  undefined  in  space,  may  yet  never  be 
disregarded  as  the  natural  starting  point  for  all  discussion. 


CHAPTER  VI. 

THE  GENERAL  NATURE  OF  MECHANICAL  ENERGY. 

The  definition  of  mechanical  energy  is  now  complete,  so  far 
as  a  definite,  though  skeleton-like,  structure  is  concerned.  But 
this  skeleton  needs  clothing  with  some  flesh  and  form,  before  it 
may  be  useful  for  a  display  of  the  nature  of  heat. 

The  outline  of  the  skeleton,  to  summarize  for  convenience, 
may  be  stated  as  follows : 

1.  Energy  has  been  identified  as  always  consisting  of  the 
arithmetical  product  of  two  variables.     One  of  these  variables 
has  been  named  the  intensity,  and  the  other  the  extensity,  of 
energy. 

2.  Intensity  has  been  shown  to  be  a  function  of  either  the 
.y/>ac£-relationship   or    the   ra  of  fora-relationship   between   two    or 
more  mass-portions.     The  intensity  of  space-relationship  is  pro- 
portional to  the  "propinquity,"  or  the  reciprocal  of  the  distance 
of  separation.     The  intensity  of  motion-relationship  is  propor- 
tional to  velocity-squared-divided-by-aggregate-mass-involved. 

3.  Extensity   has   been   shown  to  be   the   measure   of   the 
amount  of  mass-pairing  involved.    It  is  proportional,  other  things 
being  equal,  to  the  square  of  the  total  mass  involved.     For  any 
given  total  mass  it  increases,  but  not  proportionally,  with  the 
degree  of  subdivision  of  that  mass,  into  mass-pairs  capable  of 
embodying  the  relationships  defined  above. 

4.  It  was  shown  that  energy-quantities  may  vary  by  varia- 
tions in  either  intensity  or  extent.     In  the  applied  mechanics  of 
engineering  it  is  only   the  intensity-factor  which  varies   appre- 
ciably; that  is  to  say,  we  vary  space  or  motion,  while  the  mass- 
factor  remains  proportional  to  the  mass  involved.    But  in  thermal 
energy  or  other  intricate  forms,  when  viewed  mechanically,  the 
extensity-factor  must  be  expected  to  vary  as  often  and  widely 
as  the  intensity ;  that  is  to  say,  the  energy,  its  intensity  being 
fixed,  is  no  longer  necessarily  proportional  to  the  mass  involved. 

5.  The  variation  of  any  mass-system  in  intensity  of  energy 
may  take  place  smoothly,  in  stable  equilibrium,  within  a  certain 

78 


GENERAL  NATURE  OF  ENERGY       79 

range.  This  range  is  defined  at  either  end  by  the  two  critical 
intensities.  Trespass  over  either  critical  intensity  causes  the 
equilibrium  to  become  unstable.  Trespass  over  the  lower,  or 

£ 

spacial,  critical  limit  of  propinquity,  or  — ,  leads  to  collision. 

z 

Trespass  over  the  upper,  or  kinetic,  critical  limit  of  intensity,  or 

V2 

,  leads  to  dissociation.     Either  collision  or  dissociation 

Mt  +  M2 

constitutes  a  transformation  of  energy. 

6.  Mechanical  energy  existing  in  stable  equilibrium,  between 
the  critical  limits  of  intensity,  may  be  called  "permanent"  in 
form.     That  embodying  intensity  greater  than  the  spacial  limit 
of  propinquity  has  been  called  "subpermanent"  in  type.     That 
embodying  motion  above  the  upper  critical  limit  of  kinetic  in- 
tensity has  been  called  "superpermanent."     Of  the  latter  two, 
the  first  type  can  exist  only  throughout  a  portion  of  a  single 
revolution,  and  is  perceptible  to  the  human  senses  only  when  the 
members  of  the  mass-pair  are  large  enough,  and  the  period  of 
revolution  long  enough,  for  their  separate  observation.     Such  is 
the   case   in   celestial   mechanical    energies,   and   in   the   applied 
mechanics  of  machines. 

7.  It  is  next  to  be  pointed  out — and  this  is  one  of  the  most 
important  steps  in  the  understanding  of  energy — that  the  ener- 
getic conditions  of  matter,  whether  spacial  or  kinetic,  whether 
referring  to  intensity  or  extensity,  never  spring  from,  nor  are 
measurable  from,  an  absolute  zero  of  any  one  of  the  factors 
involved.     Instead,  the  factors  in  any  energetic  condition  vary 
on  either  side  of  a  central,  or  mean  energetic,  value;  and  this 
value  itself  hangs  self-supported  in  space,  so  to  speak,  with  no 
means  known  for  referring  it  to  any  absolute  base.     It  is  to  an 
explanation  of  these  statements  that  Figs.  6  and  7,  and  the  next 
few  paragraphs  of  discussion,  are  to  be  devoted. 

The   earlier   papers  of   this   series   defined   the   intensity   of 

energy  as  proportional  to    -= =-,'   when    potential,    and    of 

b0        b 

y2 y  2 

^T —       °   when  kinetic.    The  law  of  the  conservation  of  energy 

links  these  two  forms,  so  that  either  may  be  studied  as  a  repre- 
sentative of  both. 

Energy  may  consist  either  of  little  space  and  much  motion 


80  ENERGY 

(or  force),  or  of  much  space  and  little  motion  (or  force).  But 
these  paired  quantities  appear,  not  as  a  sum,  but  as  a  product. 
If  they  appeared  as  a  sum,  either  of  them  could  be  reduced  to 
zero  at  times  (the  quantity  of  energy  remaining  constant,  accord- 
ing to  the  conservation  of  energy)  by  a  sufficient  growth  of 
the  other.  But  being  bound  together  as  a  product,  neither  factor 
may  be  reduced  to  zero  by  any  Unite  growth,  however  great,  of 
the  other.  And  since  "infinities"  apply  only  to  portions  of  the 
universe  so  large  as  to  exceed  human  understanding  and  meas- 
urement, and  therefore  have  no  place  in  any  exact  natural  sci- 
ence, zeros  are  likewise  excluded  from  participation  in  energetic 
phenomena. 

Thus,    as    space    disappears,    in    nature,    energy    of    motion 

appears  in  proportion  to  ,  and  force  appears  in  proportion  to 

space 

— - —  j.    Therefore  no  finite  accumulation  of  energy  or  force, 

space  J 

however  great,  can  ever  make  the  space  zero,  or  compress  matter 
into  nothingness.  This  agrees  with  our  most  ordinary  concepts 
of  matter ;  for  two  of  the  prime  attributes  of  matter,  defined  as 
elementary  in  the  earliest  study  of  nature  as  an  exact  science, 
were  its  indestructibility  and  its  occupancy  of  space. 

On  the  other  hand,  as  space  appears  force  disappears,  and 
energy  is  absorbed.  Yet  no  imaginable  degree  of  finite  space 
can  ever  reduce  the  force  to  zero,  or  quite  annul  the  absorption 
of  energy  with  further  increase  of  space.  This  concept  dates 
from  Newton's  discovery  of  the  law  of  gravitation,  and  lies  at 
the  heart  of  our  modern  concept  of  the  universe  as  a  unit — its 
every  part  bound  inseparably  to  its  every  other  by  an  unbreak- 
able, albeit  a  very  elastic,  bond.  This  concept  is  most  familiar 
to  engineers  in  connection  with  gases  and  vapors,  which  expand 
indefinitely,  losing  pressure  as  they  go,  yet  with  no  possibility  of 
the  pressure  ever  reaching  zero. 

As  for  velocities,  they  must  follow  the  same  general  law, 
although  in  accordance  with  a  different  mathematical  function. 
Whereas  force  is  proportional  to  the  square,  and  energy  to  the 
first  power,  of  the  propinquity,  or  the  reciprocal  of  space, 
velocity  is  proportional  to  the  square  root  of  that  same  function. 
Although  the  rates  of  variation  would  therefore  differ  in  these 
three  cases,  yet  the  general  form  of  the  relationship  remains  the 


GENERAL  NATURE  OF  ENERGY 


81 


same.  No  velocity  can  be  so  great  as  to  reduce  the  space  to 
zero,  and  no  space  so  great  as  to  reduce  the  balancing  velocity 
to  zero. 

This  general  form  of  relationship  between  force,  energy  or 
motion,  on  the  one  hand,  and  space  on  the  other,  is  shown  in 
Fig.  6.  The  curve  would  not  be  the  same  in  the  three  cases,  but 
it  would  have  the  same  general  form;  and  because  of  the  diffi- 
culty of  making  one  scale  show  all  three  functions  to  advantage, 
one  only  is  shown  to  represent  them  all.  The  function  is  seen 
to  be  a  curve  asymptotic  to  the  two  rectangular  axes.  Each 
factor  may  vary  as  widely  as  it  pleases,  and  may  thereby  vary 
the  other.  But  neither  can  ever  force  the  other  to  zero,  by 
increasing  ever  so  widely. 

S0 


COMPRESSION 


FORCE  AND  MOTtO* 


FIG.  6. 

Moreover,  as  either  may  seek  to  force  the  other  to  increase, 
by  itself  decreasing  toward  zero,  it  will  find  itself  working 
against  an  increasing  mechanical  disadvantage  as  it  proceeds. 
The  further  it  goes  the  greater  is  the  proportion  of  resultant  to 
creative  action.  By  the  principle  of-  virtual  velocities,  further 
progress  must  become  more  and  more  difficult  with  each  advance. 
The  tendency  is  always  to  return  toward  a  central  or  medium 
value  for  each  of  the  factors.  When  space  becomes  deficient 
and  the  force  excessive,  force  tends  to  control  the  situation ;  as 
a  compressed  spring  or  gas  tends  to  burst  its  bonds.  When 


82  ENERGY 

space  becomes  excessive  and  force  deficient,  space  tends  to  rule 
the  game;  as  when  an  elevated  weight  tends  to  fall,  or  a  dis- 
tended gas  to  be  condensed  by  external  pressure.  The  natural 
equilibrium  in  which  these  factors  vibrate  on  either  side  of  their 
central,  or  mean  energetic,  values,  in  either  direction,  is  thor- 
oughly stable. 

This  same  general  form  of  energetic  relationship  and  sta- 
bility of  equilibrium  applies  also  to  the  other  energetic  variables, 
as  well  as  to  space,  force  and  motion.  Thus,  in  Fig.  7  can  be 
seen  the  way  in  which  the  extensity-factor,  or  quantity-factor, 
of  energy  varies  in  terms  of  the  degree  of  subdivision  of  the 
aggregate  mass  embodying  energy  of  any  stated  intensity.  In 
the  Fourth  Paper  was  developed  the  equation  for  this  relation- 
ship, in  Equation  26,  which  is  repeated  here 

X  =  W(1— i-)  (26) 

for  convenience.  In  it  X  is  the  extent  of  mass-pairing,  or  ex- 
tensity  of  energy,  which  is  embodied  in  the  mass  M  by  its  sub- 
division into  any  number  n  of  equal  parts. 

The  variation  of  X  with  n,  for  any  given  mass,  is  seen  in 
Fig.  7.  When  n  =  I,  or  the  mass  is  a  homogeneous,  solid  unit, 
embodying  one  arbitrary  unit  of  mass,  X— o  and  the  function 
appears  at  A,  Fig.  7.  When  11=2,  X— JM2  and  the  curve 
passes  to  D,  upon  a  scale  determined  by  the  size  of  the  mass- 
system  or  the  arbitrary  mass-unit  in  question.  But  as  n 
increases  still  further,  X  exhibits  an  increasing  slowness  in  fol- 
lowing proportionality  to  it.  A  doubling  of  n  to  4  increases  X 
by  only  one-half.  A  quadrupling  of  n  to  8  increases  X  by  only 
three-quarters ;  and  no  finite  extension  of  the  value  of  n,  however 
great,  can  quite  succeed  in  doubling  the  value  of  X  from  the 
point  D. 

When  the  arbitrary  unit  of  mass  which  forms  the  measure 
of  each  "equal"  portion  becomes  greater  than  one-half  the  total 
mass — that  is,  when  the  aggregate  mass  is  divided  into  only  two 
nwequal  portions,  the  larger  one  of  them  constituting  the  unit  of 
mass  and  the  fractional  remainder — n  may  have  values  (always 
positive)  which  are  less  than  two.  Such  would  be  the  case  in 
engineering  mechanics,  where  from  the  total  mass  of  the  earth 
only  a  small  fraction  is  split  off,  made  into  a  hammer  or  a 
cannon-ball  or  a  locomotive,  and  its  energy  relatively  to  the 


GENERAL  NATURE  OF  ENERGY 


83 


remainder  (which  we  still  call  "the  earth")  utilized  for  human 
purposes.  In  such  case  X  would  become  a  small  fraction 
of  M2,  and  the  curve  of  Fig.  7  would  pass  from  D  toward  C. 
When  the  total  mass  present  amounts  to  just  one  unit  of  mass, 
X  becomes  equal  to  zero.  When  the  total  mass  present  is  less 
than  one  unit  of  mass,  n  becomes  less  than  unity  and  X  becomes 
negative. 

As  the  left-hand  limb  of  the  curve  approaches  the  condition 
of  a  straight  line  parallel  with  the  axis,  the  degree  of  mass- 
pairing,  or  the  extensity  of  energy,  approaches  proportionality 
with  the  mass  of  the  smaller  fragment.  Absolute  proportionality 
is  what  is  assumed  in  the  equations  employed  in  engineering. 


-X 


EXTENT  OF  MASS-PAIRING 


XM1     -I-  X 


FIG.   7. 


But  it  is  plain  from  Fig.  7  that  this  assumption  could  become 
true  only  in  the  impossible  case  when  n  became  zero,  when  the 
fragment  split  off  from  the  earth  became  zero  and  the  extensity 
of  energy  became  minus  infinity — for  only  then  would  the  limb 
AC  of  the  curve  BAG  have  become  a  straight  line. 

It  therefore  becomes  plain  that  the  extensity  of  a  mass- 
system,  or  its  capacity  for  embodying  intensity  of  energy,  varies, 
with  the  fineness  of  its  subdivision  into  separate  portions,  quite 
as  does  space  with  motion.  The  relationship  swings  on  either 
side  of  a  central,  or  mean  energetic,  condition,  or  arbitrary  zero, 
such  as  D,  Fig.  7.  No  absolute  zero  is  attainable  in  either 
direction.  Even  if  the  axes  which  may  be  said  to  measure 
absolute  zeros,  at  the  foot  and  right-hand,  respectively,  be 


84  ENERGY 

regarded  as  basis  of  convenience  which  it  would  be  well  to  retain, 
the  fact  constantly  to  be  kept  in  mind  is  that  the  energetic 
condition  never  passes  to  either  of  them,  and  never  can.  This 
general  characteristic  holds  true,  whether  expressed  in  terms  of 
intensity  or  extensity  of  energy,  whether  of  space,  motion,  force, 
degree  of  massive  solidarity  on  the  one  hand,  or  of  fineness  of 
comminution  of  mass  on  the  other.  Any  of  these  factors  may 
pass  to  either  very  great  or  very  small  values,  but  none  of  them 
may  ever  attain  to  either  zero  or  infinity. 

Indeed,  it  will  appear,  as  the  argument  proceeds,  that  every 
energetic  relationship  which  can  be  stated  exactly  follows  this 
same  general  law.  To  those  engaged  in  power-engineering  the 
most  familiar  illustration  of  this  statement  is  the  hyperbolic 
relation  between  the  pressure  and  volume  of  any  gas.  Pressure 
and  volume  always  appear  as  a  product,  each  being  inversely 
proportional  to  the  other,  or  to  some  power  of  the  other.  The 
general  equation  is  PVx=a  constant.  They  never  appear  as  a 
sum,  one  decreasing  as  the  other  increases.  No  degree  of  pressure, 
however  great,  can  ever  reduce  the  volume  of  any  gas  to  zero; 
nor  can  any  degree  of  expansion,  however  great,  ever  reduce  the 
pressure  to  zero.  There  is  no  place  in  the  universe  where  the 
pressure  or  density  or  volume  of  elastic  matter  is  imaginably  zero. 

In  every  case,  all  energetic  functions  are  founded  upon  a 
central,  or  mean  energetic,  condition,  which  hangs  unsupported 
in  space,  so  to  speak,  as  the  sun  hangs  in  the  heavens.  No  abso- 
lute base  or  support  for  it  is  imaginable  or  necessary.  It  is  on 
either  side  of  this  central  point  of  reference,  and  not  up  and 
down  from  any  absolute  zero,  that  all  energetic  factors  vary. 
These  statements,  which  have  been  made  in  reference  to 
mechanical  energy  only,  will  be  found  to  apply  universally. 

Energetic  Equilibrium.  In  all  natural  phenomena  the  one 
most  important  guiding  principle,  after  conservation,  is  that  of 
universal  stability  of  equilibrium.  The  determination  of  what 
shall  be  the  next  in  that  most  intricate  series  of  occurrences  to 
which  we  give  the  general  name,  the  progress  of  events,  always 
depends  upon  stability  of  equilibrium.  The  natural  universe  is 
always,  except  locally  and  temporarily,  in  stable  equilibrium. 
And  if  its  equilibrium  temporarily  and  locally  has  become  un- 
stable, the  movement  is  always  toward  the  recovery  of  stability. 


GENERAL  NATURE  OF  ENERGY  85 

Whatever  may  occur  in  the  nature  of  a  departure  from  the 
general  medial  trend  of  affairs  always  brings  with  it,  as  its 
immediate  consequence,  a  tendency  to  departure  in  the  counter- 
vailing or  balancing  direction.  Although  this  tendency  may  not 
prevail  immediately,  it  must  ultimately. 

This  law  has  its  foundation  in  these  elementary  mechanical 
systems  now  under  discussion.  They  were  likened,  in  the  open- 
ing pages  of  the  second  paper,  to  the  familiar  pendulum,  which 
is  seen  to  swing  always  in  stable  equilibrium.  Turning  from 
that  to  the  less  familiar,  but  only  true,  energetic  element,  the 
two-part  free  mass-pair,  the  same  truth  appears.  The  element 
swings  in  stable  equilibrium  between  two  extremes,  one  of  un- 
usual space  and  the  other  of  unusual  force  or  motion.  In  this 
swing,  departure  in  either  direction  begets  increasingly  a  ten- 
dency to  return.  The  attainment  of  unusual  space  kills  the 
motion  which  begot  it,  and  increasingly  invites  motion  of  return. 
Unusual  lack  of  space  begets  both  force  and  velocity,  and  tends 
increasingly  to  a  reversal  of  motion  and  a  recreation  of  space. 

The  same  law  applies  to  the  mass-pairing  factor  of  energy. 
It  varies  on  either  side  of  a  central  value,  of  a  mean  average 
size  of  solid  or  undivided  mass-portion,  in  stable  equilibrium. 
The  unusual  consolidation  of  any  number  of  the  mass-portions 
of  a  system  begets  unusual  disgregative  velocity  in  the 
remainder.  This  of  itself  constitutes  a  dispersion  of  matter.  But 
in  addition,  the  unusual  velocity  of  this  remainder,  returning  in 
due  time,  tends  to  beget  a  renewed  separation  of  the  originally 
consolidated  group.  As  much  as  this  can  be  seen  in  pure 
mechanics,  with  collision  and  heat-formation  excluded  from  the 
discussion;  but  so  soon  as  these  phenomena  are  admitted,  as 
will  be  done  in  the  next  chapter,  the  field  of  this  form  of  stability 
of  equilibrium  will  reveal  its  extension  into  other  forms  of 
energy,  in  a  most  beautiful  way. 

The  same  law  applies  to  the  eccentricity  of  orbit.  Unusual 
eccentricity  tends  to  impart  energy  radially  from  the  system  to 
outside  bodies ;  and  the  loss  of  this  energy  tends  to  a  reduction 
of  the  eccentricity.  Unusual  lack  of  eccentricity,  on  the  other 
hand,  invites  the  absorption  of  energy  contributed  radially  from 
other  systems ;  and  the  effect  of  such  absorption  is  necessarily  to 
increase  the  eccentricity. 

Obviously,  too,  this  phenomenon  cannot  proceed  to  any  rigid 


86  ENERGY 

or  abrupt  limits.  Eccentricity  of  orbit  can  never  reduce  itself 
quite  to  zero,  by  the  radiation  of  energy;  because  the  ability  to 
do  so  depends  upon  the  presence  of  the  eccentricity  itself.  Cir- 
cularity of  orbit  cannot  absorb  radial  energy  to  the  point  where 
the  eccentricity  is  infinite,  because  the  ability  to  absorb  is  lost 
as  the  eccentricity  increases. 

The  fundamental  law  of  this  equilibrium,  as  evinced  between 
eccentricity,  mass  and  dimension  of  orbit,  is  based  upon  the 
conditions  found  to  prevail  in  our  solar  system;  wherein  the 
few  score  bodies,  the  motions  of  which  we  can  study,  have  had 
time,  since  the  dawn  of  astronomy  at  least,  to  settle  into  stable 
equilibrium.  The  existing  state  of  affairs  is  denned  in  the 
equation  derived  independently  by  La  Place  and  Lagrange,  viz : 

M  M 

-14pi;.e2V-STsr  =  a  constant*  (28) 

This  is  not  an  equation  of  energy-interchange,  but  one  of  fact, 
showing  the  effect  of  centuries  of  energy-interchange.  It  is 
incidental  to  and  illustrative  of  our  argument,  rather  than  basic 
for  it.  But  it  is  of  more  than  incidental  significance  that  this 
equation  reveals  the  same  general  relationship  between  the 
factors  of  energy  as  those  given  previously.  The  three  factors 
of  mass,  space  and  eccentricity  are  linked  together,  not  as  a 
constant  sum  or  a  constant  ratio,  but  as  a  constant  product. 
Any  one  of  the  three  remaining  constant  for  the  time,  either  of 
the  other  two  can  vary  to  an  unlimited  degree;  but  only  as  the 
reciprocal  of  the  third.  Neither  can  be  brought  to  zero  by  any 
expansion,  however  great,  of  the  other.  Neither  can  approach 
zero  without  encountering  increasing  resistance,  in  the  unusual 
expansion  of  the  other  which  must  accompany  it. 

In  all  of  these  respects,  the  elementary  free  mass-pair  does 
not  constitute  for  engineering  students  a  forcible  illustration,  for 
here  on  the  earth's  surface  we  have  no  free  mass-pairs  big 
enough  to  be  seen.  All  that  we  have  which  are  free  are  of 
molecular  dimensions,  and  our  knowledge  concerning  them  is 
chiefly  inference.  But  as  consideration  turns  to  the  forms  of 
energy  other  than  mechanical,  it  will  appear  that  all  energetic 

*The  writer  is  uncertain  whether  the  first  factor  in  this  equation 
should  be  as  printed,  or  simply  M.  Consistency  with  all  the  other  true 
equations  of  mechanics  would  give  it  the  form  here  printed.  But,  like 
all  these  other  basic  equations,  it  is  to  be  found  under  the  cognisance  of 
high  authorities  in  terms  of  simple  M. 


GENERAL  NATURE  OF  ENERGY       87 

action  follows,  in  a  most  striking  way,  these  same  general  char- 
acteristics. All  of  them  swing  constantly  on  either  side  of  a 
mean  energetic  condition,  against  resistances  which  increase  as 
the  departure  from  the  mean  condition  increases,  between  limits 
of  zero  and  infinity  neither  of  which  can  ever  imaginably  be 
reached.  It  is  of  vital  importance  to  the  mechanical  theories  of 
heat  and  these  other  energy-forms,  therefore,  that  these  same 
basic  characteristics  be  noied  as  attributes  of  the  most  ele- 
mentary, mechanically  energetic  mass-pair. 

Energy-transformation.  So  far  as  the  mathematical  forms 
of  the  curves  connecting  the  several  factors  of  energy  are  con- 
cerned, these  swings  of  energetic  condition  on  either  side  of  the 
central  mean  might  extend  indefinitely,  along  the  asymptotes  to 
either  axis.  But  when  mathematics  is  replaced  by  observation 
of  natural  fact,  it  appears  that  each  curve  fails  of  continuity,  if 
pushed  too  far  along  its  asymptote.  Some  factor  hitherto  irrele- 
vant enters  and  controls  the  situation.  The  energetic  equilibrium, 
stable  up  to  this  point,  becomes  abruptly  unstable.  Smooth  inter- 
action at  a  distance  between  the  two  mass-portions  comes  to  an 
end.  Either  dissociation  enters,  to  put  an  end  to  the  identity  of 
the  mass-pair  as  a  perceptible  pair,  or  collision  enters  to  put  an 
end  to  the  conservation  of  the  original  form  of  energy. 

This,  then,  is  energy-transformation,  the  break  in  the  con- 
tinuity of  the  curves  of  stable  equilibrium  and  of  visible  con- 
servation of  energy  which  reveal  the  critical  limits  of  intensity 
of  energy — the  critical  limits  to  the  concentration  of  energy  in,  or  • 
of  abstraction  of  energy  from,  a  mass-system  of  the  particular 
degrees  of  mass  and  of  mass-pairing  in  question. 

What  ensues  then  is  more  difficult  to  explain  than  what  has 
preceded.  We  know  now,  from  considerations  broader  than  any 
yet  permitted  to  enter  the  argument,  that  trespass  beyond  these 
critical  limits  of  intensity  abrogates  neither  the  Conservation  of 
Energy  nor  the  universal  Stability  of  Equilibrium.  It  is  only  in 
terms  of  the  particular  form  of  energy  in  question — in  this  case 
mechanical  energy — that  the  continuity  of  conservation  and  sta- 
bility is  broken.  The  line  is  then  crossed  which  arbitrarily 
defines  this  energy-form  from  the  others ;  and  across  this  line,  with 
the  energy,  we  must  step,  if  we  are  to  follow  clearly  the  con- 
tinuity of  universal  natural  action.  When  we  have  crossed  we 
shall  see  that  what  we  have  crossed  was  indeed  an  arbitrary  line, 


88  ENERGY 

like  a  state  boundary-line,  erected  in  the  human  imagination  to 
serve  the  convenience  of  human  limitations ;  but  having  no 
other  real  existence.  We  shall  see  that,  so  far  as  light  now 
penetrates,  all  energies  are  one  in  their  fundamental  components. 
The  sole  trouble  is  that  the  light  does  not  penetrate  far.  In 
other  energy-forms  than  mechanical  we  cannot  see  these  com- 
ponent parts.  We  know  these  more  obscure  energies  only  by 
their  blanket  results.  Yet  if  we  are  ever  to  get  any  more  clear 
and  concrete  idea  of  their  anatomy,  it  seems  inevitable  that  it 
should  be  in  terms  of  mechanical  energy.  Certainly  the  engineer 
and  the  engineering-student,  if  not  all  others,  can  proceed  more 
clearly  from  concepts  based  upon  mass,  space,  force  and  motion 
than  they  can  by  relying  solely  upon  abstract  empiricisms,  stated 
mathematically.  The  writer  will  attempt  no  argument  that  heat 
is  or  is  not  a  "mode  of  motion."  He  believes  that  it  is.  The 
evidence  that  it  is,  albeit  vague  and  inconclusive  to  some  minds, 
is  too  great  in  volume  for  denial.  We  should  lose  too  great  a 
portion  of  our  scientific  perceptions  if  we  should  deny  the  me- 
chanical nature  of  heat,  and  do  it  consistently.  But  that  is  not 
the  point.  The  point  is  that  if  heat  be  truly  a  "mode  of  motion" 
and  chemical  energy  truly  a  "mode  of  arrangement"  of  mass  in 
energetic  action,  our  concepts  thereof  must  be  guided  by  the 
fundamental  principles  of  mechanical  energy  which  have  been 
displayed  above.  We  are  now,  for  the  first  time  in  the  argument, 
mechanically  equipped  for  an  accurate  pursuit  of  the  questions: 
What  mode  of  motion?  What  mode  of  arrangement?  among 
the  many  imaginable  ones,  may  or  must  they  be  ? 


CHAPTER  VII. 

WHAT  is  HEAT? 

In  asking  ourselves  what  is  heat  the  most  surprising  thing  is 
to  think  that  not  every  one,  that  possibly  no  one,  knows  what 
heat  is.  Heat  is  one  of  the  commonest  things  in  the  world.  It 
is  as  common  as  matter ;  for  we  know  of  no  matter  without  heat. 
It  is  as  common  as  space;  for  while  space  cannot  embody  heat, 
yet  no  space  is  known  which  does  not  contain  either  matter  or 
else  that  radiant  energy  (commonly  called  "light")  which,  trav- 
eling always  at  the  inconceivable  rate  of  186,000  miles  per 
second,  turns  into  heat  as  soon  as  it  meets  solid  matter. 

Yet  we  have  no  definition  of  heat.  Heat  seems  to  be  many 
different  things,  according  to  how  it  is  encountered.  It  has  been 
called  "the  waste-heap  of  the  universe."  Indeed,  it  seems  to  be 
easier  to  say  what  heat  is  not,  than  what  itn  is.  Heat  is  not 
matter.  That  point  was  settled  a  century  ago.  >  It  is  a  "form  of 
energy."  But  all  that  that  means  is  that  it  is  capable  of  per- 
forming work.  But  as  it  is  capable  of  many  other  things  besides 
performing  work,  and  as  many  other  things  besides  heat  are 
capable  of  performing  work,  this  is  not  a  very  satisfactory 
definition. 

Heat  is  like  an  ant-colony.  It  lives  in  a  hidden  nest.  We  can 
see  it  go  into  and  come  out  of  its  nest — matter — by  several 
doors.  But  as  to  just  whither  those  doors  may  lead,  and  what 
may  be  the  form  of  structure  which  connects  the  several  doors, 
is  as  yet  pure  surmise. 

Fortunately,  there  are  quite  a  number  of  doors  to  the  thermal 
ant-hill ;  and  as  heat  appears  at  each  of  these  it  bears  a  different 
guise.  So  that,  aided  rather  than  hindered  by  the  very  diversity 
of  the  problem,  we  are  able  to  guess  fairly  near  to  the  sort  of 
interior  arrangement  which  alone  could  fit  all  of  the  doors. 

Heat  appears  in  and  disappears  from  matter  by  the  following 
processes : 

89 


90  ENERGY 

Methods    of   Heat-gain.  Methods  of  Heat-loss. 

(1)  Conduction  from  hotter  bodies.       Conduction  to  colder  bodies. 

(2)  Absorption    of    radiation.  Radiation  to  colder  bodies. 

(3)  Impact   and   friction. 

(4)  Compression.  Expansion. 

(5)  Combustion.  Dissociation. 

(6)  Electrical  resistance.  Electrical    generation    (by   thermo- 

pile). 

There  are  other  thermal  processes  than  these,  but  they  occur 
upon  too  small  a  scale  to  be  of  present  interest. 

Of  all  of  the  above  processes  the  two  most  familiar  sources 
of  heat  are  radiation  (sun-heat)  and  combustion.  But  both  of 
these  processes  are  complex  and  obscure,  when  viewed  from  the 
stand-point  of  the  present  articles,  which  are  to  concern  them- 
selves with  a  mechanical  explanation  of  heat.  Sun-heat  is  a 
transformation  of  radiant  energy,  and  combustion  a  transforma- 
tion of  chemical  energy,  into  heat;  and  both  radiant  energy  and 
chemical  energy  are  just  as  much  in  need  of  an  explanation  as  is 
heat  itself. 

Mechanical  work  is  the  only  form  of  energy  of  which  we 
now  have  any  definite  and  clear  concept.  It  is  by  the  doors 
opening  between  that  form  and  heat  that  the  latter  must  be 
approached.  These  doors  are  the  processes  numbered  three  and 
four;  and  of  these  Number  Three  comes  first,  both  numerically 
and  naturally. 

But  the  question  of  impact  and  friction  can  be  broached  for 
discussion  only  in  terms  of  elasticity  and  its  opposite. 

Elasticity  and  Inelasticity.  When  two  solid  bodies  come 
into  contact  the  collision  is  always  partially  elastic  and  partially 
inelastic.  That  is  to  say,  a  part  of  the  kinetic  energy  inherent 
in  the  bodies  before  collision  is  returned,  in  the  form  of  motion 
in  the  reverse  direction,  and  a  part  is  not.  In  so  far  as  the 
energy  is  returned  kinetically,  the  bodies  are  said  to  be  elastic. 
In  so  far  as  it  is  not,  they  are  said  to  be  inelastic.  While  some 
bodies  are  almost  perfectly  elastic,  and  others  almost  wholly 
inelastic,  none  are  known  which  are  completely  either. 

In  so  far  as  bodies  are  elastic,  their  collision  can  have  no 
effect  upon  the  mechanical  principles  laid  down  in  the  preceding 
papers.  Two  bodies  engaged  in  a  mutual  orbit  which  brought 
them  into  collision  would,  if  perfectly  elastic,  rebound  with  a 
velocity  as  great  as  that  before  collision.  Only  the  direction  of 
motion  would  be  altered.  The  original  conic-section  orbit  would 


WHAT  IS  HEAT?  91 

be  continued  unaltered,  except  that  its  new  axis  would  be 
inclined  with  its  old  one.  The  mean  energetic  and  critical  condi- 
tions would  remain  at  the  same  intensities;  but  the  collision 
which  constituted  the  lower  critical  intensity  would  not  lead  to  a 
transformation  of  energy,  as  when  inelasticity  is  present. 

Elasticity,  however,  while  incapable  of  throwing  any  light 
upon  the  principles  of  motion,  throws  considerable  light  upon 
the  energetic  nature  of  mass — at  least,  in  the  form  of  the  so- 
called  "solid"  bodies.  For  elastic  collision  means  the  temporary 
storage  of  the  energy  of  collision  in  the  deformation  of  the 
bodies,  against  their  disposition  to  retain  their  solid  form ;  which 
stored  energy  is  given  out  again  in  the  rebound,  as  the  original 
forms  are  regained. 

But,  if  our  ideas  are  to  remain  true  to  the  elements  of 
mechanical  action  as  stated  by  Kepler,  Newton  and  La  Place,  as 
collocated  in  the  preceding  papers,  this  temporary  storage  of 
energy  within  each  body  cannot  be  attributed  to  pure  mass.  It 
is  only  in  changed  relationships  between  mass-portions  that 
energy  can  be  stored.  Each  of  the  elastic  colliding  solids  must 
therefore  be  regarded,  not  as  a  unified  or  truly  solid  portion 
of  mass,  but  as  a  more  or  less  complex  system  of  mass-portions, 
between  which  the  energy  may  be  stored.  The  way  in  which 
this  may  be  done  is  not  the  point  of  immediate  interest.  The 
significant  fact  is  that  no  body  which  exhibits  any  elasticity 
whatever,  may  be  regarded  as  a  solid  unit.  No  truly  single  or 
homogeneous  body,  whether  it  be  of  the  magnitude  of  a  moon 
or  a  molecule  or  an  electron,  can  possess  elasticity,  any  more 
than  it  can  possess  energy. 

Elasticity  can  be  an  attribute  only  of  the  subdivision  of  mass. 
Wherever  the  mind  may  be  disposed  to  chase  that  most  elusive 
concept,  "the  ultimately  indivisible  portion"  of  truly  solid  or 
homogeneous  mass,  the  one  quality  which  must  be  assigned  to  it 
is  that  of  perfect  inelasticity.  The  attempted  concept  of  an 
ultimately  indivisible,  yet  perfectly  elastic,  "atom"  is  as  incon- 
sistent with  all  accurate  scientific  experience  and  principle  as  is 
the  concept  of  perpetual  motion.  Both  concepts  have  arisen 
from  the  desire  for  a  royal  road  of  unnatural  ease  to  the  solution 
of  natural  problems. 

As  for  inelasticity,  that  brings  the  discussion  home  to  the 
question  of  what  is  heat ;  for  inelasticity  is  merely  a  short  name 


92  ENERGY 

for  the  degree  to  which  kinetic  energy  is  converted  into  heat  by 
impact,  when  bodies  collide. 

In  order  to  get  the  problem  into  simple  form,  let  it  be  sup- 
posed that  the  colliding  bodies  are  perfectly  inelastic;  for  the 
addition  afterward  of  that  modicum  of  elasticity  which  is  always 
present  in  fact  will  not  affect  our  conclusions  as  to  the  portion 
which  is  inelastic.  Now  the  word  "heat"  being  merely  a  sub- 
terfuge,  or  cloak  for  ignorance,  with  which  we  cover  up  the 
fact  that  the  energy  disappears  and  we  do  not  know  what  form 
it  takes,  let  heat  be  excluded  from  the  discussion.  We  may 
refuse  to  use  the  word  until  we  have  an  exact  idea  to  attach  to  it. 

What  form,  then,  may  the  energy  of  the  colliding  bodies  take, 
if  both  elastic  rebound  and  the  formation  of  heat  are  excluded, 
and  the  conservation  of  energy  is  still  to  hold  true?  Only  one, 
by  any  possibility,  viz :  the  rupture  of  the  bodies  and  the  separa- 
tion and  scattering  of  their  fragments.  This  is  the  only  truly 
mechanical  process,  aside  from  elastic  rebound,  which  will  absorb 
energy  inelastically. 

But,  if  the  bodies  collide  as  free  bodies,  uninfluenced  by  the 
propinquity  of  other  and  greater  masses,  such  as  the  earth,  the 
fragments  will  not  stay  scattered.  They  will  fall  together  again. 
Because  of  the  increase  in  mass-pairing  by  the  splitting  up  of 
the  original  bodies,  and  also  by  the  loss  of  energy  in  their 
fracture,  the  average  intensity  of  the  fragment-pairs  must  be 
less  than  that  of  the  original  pair;  and  since  the  bodies  were  led 
to  collide,  the  fragments  will  do  likewise,  to  a  partial  degree  at 
least. 

But  these  secondary  collisions  between  the  fragments  occur 
under  the  same  conditions  as  did  the  primary  collision.  There 
is  to  be  no  elasticity,  and  no  shrouding  of  the  energy  under  the 
mysterious  term  "heat."  Therefore  the  secondary  collisions  can 
result  only  as  did  the  primary,  viz:  in  a  further  subdivision  or 
comminution  of  the  fragments.  And  these  secondary  fragments 
must  again  collide,  in  tertiary  collisions,  etc.,  etc. 

To  this  process  there  can  be  but  one  inevitable  end.  Col- 
lision, fracture  and  disgregation  must  take  place  again  and  again, 
although  with  diminishing  violence,  until  a  condition  of  per- 
manently stable  equilibrium  is  reached  by  the  fragments 
becoming  so  small  that  they  no  longer  collide.  Instead,  they  will 
have  come,  one  after  another,  as  each  became  small  enough,  to 


WHAT  IS  HEAT?  93 

adopt  elliptic  or  hyperbolic  orbits  of  revolution  about  one 
another,  without  collision,  in  permanently  stable  equilibrium  and 
with  energy  perfectly  conserved.  Their  subpermanent  energy 
will  have  become  of  the  "permanent"  type.  Their  inelasticity 
will  have  become  elastic,  not  by  some  miraculous  metamorphosis 
from  ordinary  matter  into  molecular  matter,  but  merely  by 
foregoing  contact  at  all,  procuring  reversal  of  motion  by  force- 
action  ''at  a  distance,"  instead  of  by  collision  supposed,  in 
violence  of  all  natural  experience,  to  be  perfectly  elastic. 

But  this  permanency  of  energetic  condition  is  just  what  is 
called  "heat,"  viz:  a  permanent  mode  of  motion-energy  and 
space-energy  between  the  particles  of  a  body,  resultant  from 
inelastic  collision.  The  prime  characteristic  of  heat  is  its  per- 
manence. All  other  forms  of  energy  apparently  tend  to  turn 
into  heat,  pretty  completely,  at  all  times,  while  the  heat  tends 
to  remain  heat.  That  is  why  heat  has  been  called  "the  waste- 
heap  of  the  universe,"  and  the  prediction  has  been  freely 
indulged  in  that  ultimately  all  other  energy  in  the  universe  must 
become  heat. 

The  writer  would  explicitly  avoid  giving  countenance  to  so 
extreme  a  belief  as  this.  Yet  undoubtedly  the  prime  charac- 
teristic of  heat  is  its  permanence  and  stability  of  equilibrium, 
when  considered  as  a  result  of  and  in  contrast  with  the  abrupt 
instability  of  the  mechanical  energy  of  solids  and  liquids  moving 
in  contact.  Temporarily,  at  least,  the  energy  has  reached  a 
permanence  of  form  which  must  extend  over  a  period  covering 
many  millions  of  the  vibrations  of  the  very  tiny  mass-pairs 
embodying  the  heat — far  too  long  to  permit  the  hypothesis  of 
collision  occurring.  For  there  is  no  collision  known  to  science 
which  is  not  somewhat  inelastic  and  dissipative  of  its  energy. 

The  attempt  to  define  heat  has  therefore  accomplished  its 
first  stage.  Heat  is  a  mode  of  motion  and  of  separation  among 
a  swarm  of  tiny  fragments  of  the  mass  of  the  hot  body,  none 
of  which  possess  subpermanent  orbits. 

As  to  superpermanency  of  orbit,  discussion  of  the  possibility 
of  that  being  a  part  of  thermal  interaction  must  be  deferred. 
Other  questions  as  to  the  form  of  these  intermolecular  orbits 
and  interactions  must  also  be  deferred,  until  the  many  diverse 
peculiarities  of  heat  may  have  been  gotten  more  clearly  in  view. 
To  this  end  good  use  can  be  made  of  the  thermal  diagram. 


94  ENERGY 

And  when  all  the  data  are  thus  displayed  it  may  appear  that 
several  of  the  other  sources  of  heat,  which  are  much  more 
obscure  in  form  and  nature  than  impact  and  friction,  have 
characteristics  so  like  to  these  that  this  entrance  into  the  thermal 
ant-hill  through  only  one  of  the  many  doors  may  not  seem  so 
one-sided  and  inconclusive  a  plan  after  all. 


CHAPTER  VIII. 

THE  THERMAL  DIAGRAM. 

Let  heat-producing  impact  and  friction  be  imagined  as  occur- 
ring  against  a  specific  weight,  such  as  one  pound,  of  some 
familiar  solid,  such  as  ice.  Let  it  be  imagined  that  the  ice  were 
originally  at  the  very  lowest  temperature  of  which  scientific 
investigation  has  had  experience,  where  the  ice  would  be  a  very 
cold,  hard,  brittle  solid.  The  result  of  the  impact  and  friction 
would  be  to  raise  the  temperature  of  this  solid;  and  if  it  were 
continued  sufficiently,  it  would  ultimately  melt  the  ice  and  carry 
the  resultant  water  through  all  the  thermal  experiences  of  which 
the  substance  H2O  is  capable. 

It  will  be  of  convenience  to  represent  this  process  graphically. 
And  since  the  previous  analysis  has  shown  energy  always  to 
consist  of  the  product  of  two  independent  variables,  it  will  be 
most  natural  to  regard  thermal  energy  also,  from  the  start,  as 
made  up  of  the  arithmetical  product  of  two  variables.  Indeed, 
if  heat  is  to  be  considered  as  a  "mode  of  motion,"  or  one  form 
of  mechanical  energy,  at  all,  it  must  be  considered  as  the  arith- 
metical product  of  two  variable  factors;  for  this  constitution 
was  everywhere  found  to  be  a  prime  characteristic  of  mechanical 
energy. 

When  any  quantity  thus  consists  of  the  product  of  two 
variables,  it  is  most  conveniently  represented  as  an  area,  upon  a 
field  of  rectangular  coordinates.  The  two  independent  factors 
then  become  the  two  coordinates,  respectively.  But  if  heat  is  to 
be  depicted  thus,  the  identity  of  the  two  coordinate  factors  stands, 
at  the  start,  as  a  matter  of  guess-work.  An  easy  first  guess  for 
one  of  them  is  temperature ;  for  temperature  has  been,  from  the 
beginning  of  thermal  science,  recognized  as  a  prime  factor  in 
heat.  And  yet  it  has  also  long  been  known  that  temperature  is 
not  heat. 

~  If  this  first  guess  has  been  a  true  one,  in  accord  with  the 
natural  facts,  then  the  second  coordinate  (regarding  which  noth- 
ing can  be  known  at  the  start)  will  prove  to  be  identical  with 

95 


96  ENERGY 

some  natural  prime  factor  in  thermal  phenomena  also.  But  if 
the  first  guess  should  prove  to  have  been  wrong,  then  the  whole 
graphical  situation  will  fall  into  chaos,  in  a  reductio  ad  absurdum. 

But  if  a  fair  knowledge  of  thermodynamics  on  the  part  of 
the  reader  be  assumed,  it  will  be  plain  that  the  selection  of  tem- 
perature as  one  of  the  prime  factors  of  heat  is  no  wild,  irre- 
sponsible guess.  It  is  now  more  than  eighty  years  since  Carnot 
proved  conclusively  that  temperature  was  the  one  fundamental 
feature  of  heat  in  the  guidance  of  work-performance;  and  that, 
too,  by  pure  empiricism,  without  attempting  any  definition  of 
either  heat  or  temperature.  It  is  now  more  than  a  quarter  of  a 
century  since  Lord  Kelvin  defined  the  only  true  temperature- 
scale  in  terms  of  work,  rather  than  of  heat;  and  since  Maxwell 
proved,  in  the  mathematical  theory  of  the  so-called  "perfect 
gas,"  that  temperature  was  the  translational  kinetic  energy  of 
the  flying  particles  of  thermal  matter,  and  therefore  a  real 
physical  quantity. 

But  the  writer  wishes  especially  to  avoid  humbugging  both 
himself  and  the  reader  by  starting  from  premises  which  are  laid 
down  too  rigidly,  as  if  they  were  absolute  law.  For  in  this  whole 
field  of  discussion  we  possess  no  such  rigid  premises — unless  the 
laws  of  Kepler  and  Newton,  which  are  now  unquestionable,  be 
such.  The  ideas  as  to  temperature,  heat,  entropy,  etc.,  must  fit 
the  facts;  that  is  all.  While  the  present  discussion  has  started 
rigidly  enough  from  the  exact  mechanics  of  Kepler,  Newton  and 
La  Place,  because  their  work  has  stood  the  tests  of  centuries,  yet 
the  entire  hypothesis  that  heat  is  "a  mode  of  motion"  at  all,  it 
must  be  remembered,  yet  hangs  unsettled  in  mid-air.  If  it  can 
be  made  to  appear  that  what  exact  data  we  possess  as  to  heat  fit 
what  exact  data  we  possess  as  to  mechanics,  heat  may  be  accepted 
as  a  mode  of  motion.  But  until  that  is  settled  our  premises  must 
remain  assumptions  and  guess-work,  and  should  be  defined 
clearly  as  such. 

With  this  to  start  with,  the  thermal  diagram  may  be  put 
under  construction,  as  at  B,  at  the  lower  left-hand  corner  of 
Fig.  8.  The  vertical  axis  of  this  diagram  is  to  measure  "abso- 
lute" temperature,  along  the  axis  OT.  This  locates  two  hori- 
zontal axes:  one  at  XX  for  the  absolute  zero  of  temperature, 
and  the  other  at  ZZ  for  the  Fahrenheit  zero. 

Areas,  then,  are  to  measure  heat,  and  heat  as  supplied  by 


THE  THERMAL  DIAGRAM  97 

impact  and  friction.  But  the  other  coordinate  of  the  diagram 
is  for  the  present  unknown.  Therefore  it  must  be  defined  in 
terms  of  the  two  quantities  which  are  known.  Plainly,  it  must 
be  the  result  of  dividing  area  (or  heat)  by  height  (or  tem- 
perature). 

But  in  doing  this  it  must  be  remembered,  as  was  stated  in 
the  opening  pages  of  the  First  Paper,  that  energy  is  a  name  for  a 
change  of  condition  only,  and  not  for  something  absolute.  There- 
fore, since  conditions  change  constantly  with  increments  of 
energy,  our  definition  must  be  confined  to  exceedingly  small 
increments  of  energy  at  a  time  ;  or,  in  short,  must  be  stated  as  a 
differential. 

Let  the  horizontal  coordinate  be  given  the  symbol  N.  Then 
our  stated  premises  are  defined  mathematically  by  Equation  29, 

dN--K--  .  (29) 


wherein  dQ  signifies  the  quantity  of  impactive  energy  absorbed, 
T  the  absolute  temperature  of  the  body  at  the  time  of  impact, 
and  K  the  specific  heat  of  the  body.* 

Starting  therefore  at  B,  Fig.  8,  the  curve  which  represents 
the  thermal  experiences  of  the  ice  under  impact  must  be  some 
such  an  one  as  BC,  simultaneously  rising  in  temperature  and 
departing  to  the  right,  with  positive  values  for  dN.  The  equation 
for  this  curve  can  be  had  by  integrating  Equation  29,  which  is 
easily  done  if  K  the  specific  heat  be  a  constant.  The  result 
then  is 

-  (30) 


wherein  the  zero-subscripts  refer  to  any  original,  and  the 
unmarked  symbols  to  any  final,  condition.  Should  the  specific  heat 
be  not  a  constant,  the  form  of  the  curve  would  be  slightly  altered 

*The  writer  has  no  desire  to  impose  upon  the  reader  the  suggestion, 
from  the  above  language,  that  the  thermal  diagram  thus  developed,  which 
will  prove  to  be  identical  with  the  well  known  entropy-temperature  dia- 
gram, is  originated  in  these  pages.  But  the  ultimate  significance  of  this 
diagram  has  been  so  doubtful  and  obscure  that  the  writer  wishes  to  avoid 
its  adoption  in  the  premises.  The  argument  has  been  arranged  to  be  self- 
contained.  No  dogmatic  assertions  have  been  made  in  the  premises, 
except  the  mechanics  of  Kepler  and  Newton  and  the  principles  of 
the  conservation  of  mass  and  energy.  If  the  conclusions  appear  to  be 
too  faulty,  or  too  extensive  to  be  easily  accepted,  the  blame  will  not  be 
lost  amid  a  haze  of  ill-defined  or  too  rigidly  assumed  premises. 


98  ENERGY 

and  the  integration  not  so  easy;  but  the  general  conclusions 
would  be  unaltered. 

During  this  warming  of  the  ice  by  impact  there  occurs,  appar- 
ently, no  change  in  its  physical  state.  The  ice  remains  a 
crystalline  solid.  Only  its  temperature  alters,  together  with  the 
yet  undefined  horizontal  factor  of  heat.  From  this  fact  the 
process  and  curve  BC,  with  Equation  30,  have  been  given  the 
name  isomorphic — the  syllable  "morph"  indicating  form  and  the 
prefix  "iso"  the  fact  that  it  remains  unchanged.  The  term  is 
used  in  contrast  with  metamorphic,  which  is  applied  to  those 
processes,  such  as  fusion,  vaporization,  etc.,  where  the  physical 
state  of  the  body  does  undergo  a  material  change  in  form. 

From  Equation  30  it  is  clear  that  if  the  original  temperature 
T0  from  which  the  ice  was  warmed  be  imagined  as  occurring  at 
lower  and  lower  points,  the  point  B  must  be  regarded  as  moving 
indefinitely  to  the  left,  as  it  approaches  the  "absolute  zero"  limit 
of  temperature.  The  curve  BC  must  be  asymptotic  to  this  axis, 
as  it  trends  to  the  left.  No  finite  value  of  N  —  N0  can  be  great 
enough  to  make  T0=o,  when  the  value  of  T  is  finite.  That  is 
to  say,  the  absolute  zero  of  temperature  is  absolutely  unattainable 
in  nature.  It  is  as  unreal  as  is  any  absolute  zero  of  either  space, 
motion,  force  or  mechanical  energy. 

As  the  isomorphic  BC  proceeds  to  the  right,  however,  it  finds 
an  abrupt  termination  at  the  point  C.  When  a  temperature  of 
32°  F.  is  reached  the  ice  begins  to  melt.  Further  supplies  of 
impactive  or  frictional  energy  continue  to  be  absorbed,  and  it  is 
to  be  inferred  that  they  take  the  form  of  heat.  This  fact  can  be 
easily  proven  by  other  experiments,  such  as  mixing  the  ice  with 
red-hot  iron,  when  the  iron  will  be  cooled  and  the  ice  melted. 
But  the  latter  would  not  be  warmed.  It  would  absorb  the  heat 
given  up  by  the  iron  as  "latent"  heat,  in  a  change  of  physical 
state  at  constant  temperature. 

Such  a  process  would  be  shown  in  Fig.  8  by  the  straight  hori- 
zontal line  CA.  It  is  horizontal  because  the  temperature  does 
not  change  during  the  heat-absorption.  It  is  straight  and  dotted 
because  it  does  not  represent  a  continuous  process,  but  a  gap, 
across  which  jumps  molecule  after  molecule  of  ice,  as  it  acquires 
heat  enough,  in  unstable  equilibrium. 

It  is  known  that  the  amount  of  heat  absorbed  in  this  process 
is  large  (142  B.t.u.  per  pound).  It  is  known  that  the  melting 


THE  THERMAL  DIAGRAM  99 

consists  of  a  breaking  up  of  the  ice-crystals  into  formless  liquid. 
The  volume  of  resultant  water  is  less  than  that  of  the  ice; 
nevertheless,  there  is  reason  to  believe  that  the  space  actually 
occupied  by  the  substance  itself  increases  during  fusion.  The 
ice-crystals  may  be  likened  to  a  cord  of  fire-wood ;  when  con- 
verted into  saw-dust  less  space  will  be  occupied  than  by  the  pile 
of  logs,  with  their  many  interstices;  yet  if  these  interstices  be 
deducted  from  the  original  volume,  the  space  occupied  by  the 
wood  has  increased.  A  few  large  interstices  have  been 
exchanged  for  many  small  ones — the  former  being  distinguishable 
from  the  substance,  but  the  latter  not. 

Such  a  process  as  this  melting  of  the  ice  is  called  meta- 
morphic,  implying  change  of  form,  in  contrast  with  the  isomor- 
phic  processes  such  as  BC  The  metamorphic  processes  are  all 
virtually  isothermal.  The  heat  which  they  absorb  is  called  latent, 
because  imperceptible  by  the  thermometer,  in  contrast  with  the 
sensible  heat  absorbed  in  the  isomorphic  processes. 

Equation  29  easily  becomes 

dQ=TdN  (31) 

If  the  relation  between  T  and  dN  be  known,  as  by  knowing  the 
specific  heat,  this  equation  can  be  integrated  into  an  expression 
for  Q  —  Qo,  the  quantity  of  heat  involved  in  passing  from  any 
one  condition  to  any  other.  The  quantity  dQ  is  shown  in  Fig.  8, 
as  a  very  narrow  vertical  rectangle,  having  a  height  T  and  a 
width  dN.  The  integration  of  this  rectangle  will  develop  the 
area  beneath  the  curve  bounding  the  upper  ends  and  limited  at 
right  and  left  by  the  ordinates  of  original  and  final  conditions. 
Thus,  the  heat  required  to  warm  the  ice  from  the  condition  B  to 
the  condition  C  will  be  given  by  the  area  BCc.  That  required 
to  melt  the  ice  will  be  given  by  the  area  CAOc. 

The  melted  ice  at  the  point  A  constitutes  the  arbitrary  zero 
of  our  steam-tables.  If  the  process  of  adding  energy  by  impact 
and  friction  be  continued,  the  water  will  renew  its  rise  in  tem- 
perature, this  time  along  the  isomorphic  ADD'W.  At  the  same 
time  its  vapor-tension,  or  the  pressure  which  is  exerted  at  all 
temperatures  by  the  vapor  struggling  to  free  itself  from  the 
water,  increases. 

If  this  vapor-tension  should  happen  to  equal  the  pressure 
exerted  upon  the  water  by  the  surrounding  objects  by  the  time 


100 


ENERGY 


the  temperature  D  is  reached,  the  molecular  equilibrium  again 
becomes  unstable.  The  water  cannot  absorb  more  energy  in  the 
shape  of  internal  motion  without  its  internal  centrifugal  pressure 
exceeding  the  external  centripetal  pressure;  wherefore  it  must 
burst.  So  burst  it  does,  molecule  after  molecule,  into  steam,  as 
fast  as  each  molecule  becomes  hot  enough;  just  as  pop-corns 
do  in  a  corn-popper. 

The  popping  of  each  molecule  alters  its  condition  suddenly 
and  completely  from  that  shown  at  D  to  that  shown  at  E.  There 
is  no  stability  of  equilibrium  between  D  and  E,  and  no  molecule 
may  stop  part  way  after  having  once  started  across.  If  the 
thermal  condition  of  a  quantity  of  water  and  steam  is  ever 
sliown  at  any  intermediate  point,  such  as  P  or  Y,  it  means  not 
that  the  entire  pound  of  molecules  is  in  such  intermediate  con- 
dition, but  that  one-half  or  less  are  in  the  condition  D  and  the 
other  half  or  more  are  in  the  condition  E. 


FAHRENHEIT  ZERO 


OF  TEMPERATURE 


AGSOLUTEZERO" 
OF  TEMPERATURE 


FIG.  SA. 


THE  THERMAL  DIAGRAM 


101 


The  latent  heat  of  vaporization  is  shown 
by  the  area  beneath  DE,  or  dDEe.  It  is  much 
greater  than  the  latent  heat  of  fusion,  as 
shown  by  comparing  this  area  with  CAOc. 
In  this  case  there  is  no  doubt  as  to  the 
increase  in  volume.  The  volume  of  each 
molecule  after  "popping"  is  several  hundred 
times  that  before,  the  ratio  depending  upon 
the  pressure  under  which  vaporization  occurs. 

Indeed,  this  increase  in  volume  is  so  great 
that  it  expands  into  visibility  an  energy- 
quantity  which  has  hitherto  been  negligible. 
This  is  the  external  work.  The  energy 
supplied  to  each  molecule  along  DE  con- 
sists not  only  of  that  required  to  burst 


HI 


THE  MINIMUM  LIMIT  OF  TEMPERATURE 

FIG.  SB. 


102  ENERGY 

the  molecule  against  its  own  internal  bonds  of  unity,  called  the 
disgregation~workt*  but  the  external  forces  must  be  pushed  back 
also,  throughout  a  considerable  increase  in  volume.  Yet  in  this 
case  the  external  work  amounts  only  to  about  one-ninth  to  one- 
fifteenth  of  the  disgregation-work,  so  powerful  are  the  congre- 
gative  molecular  forces. 

If  the  pressure  upon  the  heated  water  had  been  higher  than 
that  permitting  vaporization  at  D,  that  process  would  have  been 
delayed  until  some  higher  temperature  had  been  reached,  as  at 
D'.  Therefore,  there  may  be  as  many  different  vaporization- 
levels  as  there  are  different  pressures.  The  ends  of  these  various 
metamorphic  lines,  such  as  DE,  D'E',  etc.,  form  a  curve  SS  which 
is  called  the  saturation-curve.  But  the  student  is  especially 
warned  against  thinking  of  the  saturation-curve  as  representing 
a  process,  as  do  most  of  the  other  curves  in  the  thermal  diagram. 
There  is  no  process  known  to  the  boiler  or  engine  rooms  which 
will  convert  saturated  steam  of  one  pressure  and  temperature 
into  saturated  steam  of  another  pressure  and  temperature.  It 
takes  a  combination  of  at  least  two  processes  to  do  this,  and  an 
impossibly  delicate  balance  of  the  two,  at  that — unless  the  steam 
be  in  contact  with  water  with  which  it  may  interchange  heat 
promptly,  in  which  case  the  heat  is  added  not  merely  to  the 
steam,  but  to  the  water  also.f 

If  the  addition  of  energy  by  friction  and  impact  to  the  pound 
of  H2O  at  E  be  continued  still  further,  the  steam  will  rise  in 
temperature  again  along  the  isomorphic  EF  of  superheated 
steam.  The  volume  increases  almost  proportionally  with  the 
temperature.  The  disgregation-work  has  now  become  the  minor 
portion  of  the  energy  absorbed,  the  external  work  being  in 
ascendancv. 


*"Disgregation"  implies  the  scattering  of  a  flock,  the  opposite  of  "con- 
gregation," the  gathering  of  a  flock. 

fFor  this  reason  the  use  of  the  term  "specific  heat  of  saturated  steam,'* 
meaning  the  difference  in  the  total  heats  of  two  points  on  the  saturation- 
curve  separated  vertically  by  one  degree  of  temperature,  cannot  be  too 
strongly  condemned,  as  loose  and  misleading.  The  term  "specific  heat" 
is  used  generally  and  properly  to  signify  the  quantity  of  heat  which,  added 
to  a  substance,  will  raise  its  temperature  by  one  degree,  the  physical 
state  remaining  constant.  The  use  of  the  term  "snecific  heat  of  saturated 
steam"  therefore  leads  the  student  to  infer,  most  naturally,  that  tf  you 
add  to  saturated  steam  containing  the  total  heat  Hi  the  heat  H2— Hi, 
wherein  H2  is  the  total  heat  of  saturated  steam  one  desrree  hotter,  there 
would  result  saturated  steam  of  the  total  heat  H»  and  temperature  Ti+l, 
Yet  nothing  could  be  further  from  the  truth. 


THE  THERMAL  DIAGRAM  103 

At  some  higher  temperature,  such  as  F,  there  arises  a  new 
condition  of  unstable  internal  equilibrium  within  the  molecule. 
If  energy  continues  to  be  added,  molecule  after  molecule  of  the 
steam  bursts  again — but  this  time  not  into  a  new  "physical" 
state  of  H2O,  but  into  a  new  '"chemical"  state  of  dissociated 
hydrogen  and  oxygen.  The  energy  now  absorbed  in  potential 
form,  along  the  metamorph  FG,  is  some  seven  times  as  great  as 
that  absorbed  in  vaporization ;  just  as  that  absorbed  in  vaporiza- 
tion was  some  seven  times  that  involved  in  fusion.  Moreover, 
it  is  not  called  latent  thermal  energy,  but  chemical  energy.  For 
this  practise  there  are  excellent  reasons;  but  it  is  one  of  the 
offices  of  Fig.  8  to  show  plainly  how  much  more  closely  allied 
are  heat  and  chemical  energy  than  is  commonly  supposed.  It 
is  also  to  remind  the  reader  that  chemical  energy  is  distinctly  a 
latent  form  of  energy. 

The  increase  in  volume  which  is  involved  in  this  dissociation 
amounts  to  only  fifty  per  cent. ;  and  the  external  work  involved 
amounts  to  less  than  two  per  cent,  of  the  whole.  Therefore  it 
may  be  said  that  virtually  all  of  the  energy  absorbed  goes  into 
latent,  chemical  or  disgregative  form. 

Should  the  process  of  adding  energy  by  impact — which  has 
now  become  most  difficult,  because  of  the  almost  perfect  elas- 
ticity of  the  substance — be  persisted  in,  the  temperature  again 
rises  along  an  isomorph,  GH;  and  to  this  isomorph  there  is  no 
further  interruption,  by  instability  of  internal  equilibrium,  so 
far  as  the  writer  is  aware.  Moreover,  because  of  the  high 
temperature  already  attained,  the  horizontal  departure  dN  in- 
volved in  the  absorption  of  a  thermal  unit  dQ  has  become  so 
small,  and  the  corresponding  rise  in  temperature  has  become  so 
great  (because  the  specific  heat  is  much  less  than  for  liquids  or 
solids),  that  the  isomorph  GH  rapidly  becomes  approximately 
asymptotic  to  some  limiting  vertical  axis,  not  yet  exactly  located. 

Should  the  oxy-hydrogen  mixture  which  is  illustrated  as 
active  in  the  isomorph  GH  be  cooled  again,  by  the  conduction 
of  its  heat  to  colder  bodies,  it  may  be  said  (though  the  statement 
is  unsupported  by  what  has  preceded)  that  it  will  return  along 
the  curve  HGFEDACB  which  it  came  up.  But  if  the  two  gases 
be  separated,  and  then  cooled,  their  thermal  condition  will  be 
shown  by  the  curve  HGRh. 

If,    at   any  comparatively  low  temperature   such   as    h,   the 


104  ENERGY 

gases  be  mixed  again  and  then  heated,  when  they  reach  their 
temperature  of  ignition,  as  at  R,  they  will  again  combine, 
releasing  chemical  energy  which  must  be  absorbed  thermally.  In 
order  that  the  combination  may  be  complete,  the  thermal  energy 
must  be  absorbed  from  the  substance,  by  some  outside  body, 
bringing  the  mixture  again  into  the  condition  F.  The  sum  of 
these  processes  may  be  illustrated  by  the  curve  RF,  although  an 
intermediate  passage  of  some  portions  of  the  substance  into  the 
condition  H  is  involved. 

In  the  curve  hRH  is  visible  a  smooth  transit  of  a  substance 
from  one  thermal  condition  which  is  asymptotic  to  the  axis  of 
absolute  zero  of  temperature,  to  another  thermal  condition  which 
is  asymptotic  to  an  axis  at  right  angles  thereto.  In  the  curve 
BCADEFGH  is  visible  a  somewhat  similar  path  of  thermal 
transit,  broken  by  localities  of  unstable  equilibrium,  it  is  true, 
but  terminating  in  each  direction  in  a  smooth  curve  of  stable 
equilibrium,  which  is  asymptotic  to  horizontal  and  vertical  axes, 
respectively.  The  field  visible  between  these  curves  is  crossed, 
from  curve  to  curve,  back  and  forth,  by  the  familiar  processes 
of  melting,  freezing,  vaporizing,  condensing,  combustion  and 
dissociation.  The  curves  themselves  represent  the  even  more 
familiar  processes  of  warming  and  cooling. 

If  it  be  remembered  that  every  substance  known,  except  a 
few  gases  rare  even  in  the  chemical  laboratory,  occurs  in  all  of 
the  three  physical  states:  solid,  liquid  and  gaseous,  and  is  subject 
to  chemical  dissociation  and  combination,  in  endothermic  and 
exothermic  processes;  and  if  it  be  remembered  that  all  these 
processes,  for  all  these  substances,  might  be  represented  upon 
Fig.  8,  with  no  departure  from  what  is  already  there,  except  in 
the  number  and  confusion  of  lines  and  in  the  choice  of  con- 
venient scales;  and  if  it  be  further  remembered  that  all  the 
processes  for  conversions  between  heat  and  work,  as  well  as 
those  between  heat  and  chemical  energy,  may  be  displayed  upon 
this  diagram  (though  as  yet  there  has  been  no  reason  to  refer 
to  some  of  them) — it  becomes  evident  that  Fig.  8  places  before 
the  eye  in  comprehensive  form  a  pretty  complete  picture  of  the 
data  which  are  concerned  in  and  essential  to  a  complete  under- 
standing of  the  relationships  existing  between  heat,  work  and 
chemical  energy.  If  there  be  any  truth  in  the  hypothesis  that 
heat  is  indeed  a  mode  of  motion,  and- chemical  energy  a  mode 


THE  THERMAL  DIAGRAM  105 

of  arrangement,  it  should  certainly  come  out  from  a  thorough 
inspection  of  Fig.  8,  in  terms  of  the  analysis  of  mechanical 
energy  already  accomplished. 

This  task  must  be  deferred  for  later  papers.  For  the  present 
it  is  desired  merely  to  call  attention  to  the  general  fact  that  the 
curves  of  energetic  equilibrium  displayed  in  Fig.  8  have  the  same 
general  form  as  those  displayed  in  the  Sixth  Paper,  covering 
all  known  cases  in  purely  mechanical  energy.  That  is  to  say, 
they  vary,  on  either  side  of  a  central,  or  mean  energetic,  con- 
dition, which  central  condition  is  not  definitely  located  from  any 
rigid  or  absolute  base,  indefinitely  toward  and  along  two  axes 
(at  right  angles  with  each  other)  to  which  they  become  asymp- 
totic and  which  they  can  never  reach.  Unlimited  departure  from 
either  axis  is  natural  and  imaginable;  but  unlimited  approach 
toward  either  one  is  not.  While  it  is  true  that  the  upper  limb 
of  the  heat-curve  of  Fig.  8  is  not  a  true  asymptote,  as  it  is  there 
drawn,  yet  its  similarity  to  one  is  obvious.  Whether  or  not  it  be 
a  true  asymptote  will  be  discussed  more  in  detail  later. 


CHAPTER  IX. 

MECHANICAL  CONCEPTS  OF  THERMAL  PHENOMENA. 

A.     PRESSURE  AND  VOLUME. 

If  it  be  assumed  that  the  preceding  papers  have  supplied 
complete  data  for  the  understanding  of  work,  heat  and  chemical 
energy,  in  so  far  as  the  last  named  is  related  to  the  other  two, 
the  first  task  is  to  construct  therefrom  mechanical  concepts  of 
the  four  fundamental  attributes  of  matter  which  are  active  in 
these  fields,  viz : 

(a)  Pressure, 

(b)  Volume, 

(c)  Temperature,  and 

(d)  Entropy. 

These  are  all  thermal  attributes.  Pressure  and  volume  are  ther- 
modynamic  in  character,  bridging  the  gap  between  heat  and  work. 
Temperature  and  entropy  are  almost  purely  thermal  attributes, 
but  also  are  active  in  thermochemical  phenomena. 

Besides  the  above  attributes,  it  is  also  necessary  to  explain  in 
mechanical  terms,  two  fundamental  processes,  viz: — 

(e)  Heat-development  and  transfer,  and 

(f)  Thermodynamic    work-performance  or  work-absorption. 

Of  all  of  these  phenomena,  according  to  the  writer's  view, 
the  one  easiest  to  comprehend  is  volume,  and  the  hardest  one 
is  pressure.  Both  are  inextricably  mixed  up  with  heat-action; 
and  yet  the  occurrence  of  mechanical  action  where  pressure 
and  volume  are  involved,  yet  where  no  heat-changes  are  per- 
ceptible, is  common.  But  in  all  cases,  if  the  facts  be  closely 
examined,  both  pressure  and  volume  will  be  found  to  be  thermal 
phenomena.  For  no  substance  can  be  imagined  as  reduced  to  a 
zero  of  either  pressure  or  volume  without  first  being  carried  to 
that  unattainable  thermal  condition,  the  absolute  zero  of  tempera- 
ture. 

106 


MECHANICAL  CONCEPTS  107 

Volume.  If  the  idea  that  heat  is  a  mode  of  motion  is  to  be 
adhered  to  consistently,  the  volume  exhibited  by  any  body  must 
be  the  effect  of  the  separation  between  its  component  mass- 
particles.  Since  the  internal  condition  of  the  body  is  permanent 
and  stable,  these  mass-particles  must  be  in  a  free  condition  of 
natural  equilibrium.  If  so,  their  relative  separation,  in  the  face 
of  their  mutual  attraction  toward  each  other,  as  well  as  in  the 
face  of  external  pressure,  must  be  maintained  by  motion  of 
revolution  about  each  other.  This  is  the  only  possible  me- 
chanical explanation  of  the  occupancy  of  space  by  elastic  matter. 

In  order  to  explain  volume  alone,  this  internal  motion  might 
be  regarded  as  purely  circular  in  form;  and  this  is  the  simplest 
idea  with  which  to  start.  In  that  case,  the  centrifugal  and 
centripetal  forces  would  be  exactly  balanced,  and  there  would 
be  no  active  exertion  of  expansive  pressure  outwardly.  There 
would  be,  however,  a  passive,  and  at  times  a  stalwart,  resistance 
to  compressive  pressures  acting  radially  inwardly. 

Since  the  motion  of  the  particles  (under  the  above  assump- 
tion) is  purely  tangential,  its  velocity  must  increase  as  the 
volume  of  the  body  becomes  smaller.  But  during  any  such  a 
change,  where  circularity  of  motion  is  conserved,  the  energy 
cannot  be  conserved.  Energy  must  be  abstracted  in  order  that 
the  volume  of  the  body  should  become  smaller  and  the  tan- 
gential velocities  greater. 

The  only  special  provision  to  be  made,  in  imagining  the 
volume  of  a  body  as  made  up  of  a  vast  number  of  tiny  mass- 
pairs,  revolving  as  described  in  the  earlier  papers  upon  me- 
chanical energy,  is  that  these  many  orbits  must  lie  in  all  sorts 
of  planes,  interacting  at  oblique  angles, ^whereas  the  elementary 
orbits  studied  were  all  specified  as  confined  to  a  single  plane. 
But  this  provision  introduces  no  new  principles  of  action. 

Pressure.  The  above  hypothesis  furnishes  no  explanation 
of  pressure  active  outwardly.  For  this  can  be  explained  only 
upon  the  assumption  of  eccentricity  of  orbit  between  the  par- 
ticles, involving  as  it  must  a  lack  of  balance  between  centrifugal 
and  centripetal  forces  at  both  apastron  and  periastron.  Indeed, 
this  lack  of  balance  is  true  of  every  point  of  an  eccentric  orbit 
except  the  mean  energetic  point;  and  even  there,  although  the 
forces  are  balanced,  there  exists  an  unbalanced  fund  of  radial 


108  ENERGY 

motion,  acting  outwardly  on  one  side  of  the  orbit  and  inwardly 
on  the  other. 

Now  all  known  conditions  of  matter  exhibit  pressure  or  force 
per  unit  of  area.  This  pressure  is  of  the  two  sorts  already  men- 
tioned, viz :  first,  the  passive  resistance  to  external  forces  which 
is  exhibited  at  its  best  in  the  solids;  and,  secondly,  the  sponta- 
neous expansive  pressure  which  is  best  exhibited  in  the  gases. 
But  all  known  forms  of  matter  exhibit  both  of  these  phenomena. 
There  are  no  known  solids  so  dense  and  hard  that  they  do  not 
exert  some  slight  expansive  vapor-pressure,  although  the  pas- 
sively resistant  form  of  pressure  is  overwhelmingly  more  promi- 
nent. On  the  other  hand,  there  are  no  known  gases  so  diffuse 
that  they  do  not  exhibit  some  slight  resistance  to  deformation, 
called  "viscosity,"  such  as  is  familiar  in  all  liquids  and  solids, 
although  their  gaseous  characteristics  are  overwhelmingly  more 
prominent. 

Therefore,  since  both  sorts  of  pressure  are  to  be  found  in 
finite  degree  in  all  cases,  and  since  neither  sort  of  pressure  can 
be  explained  mechanically  without  finite  eccentricity  of  orbit,  it 
must  be  assumed  that  all  molecular  orbits  are  somewhat  eccen- 
tric. Neither  circular  nor  rectilinear  orbits  are  possible. 

This  supposition  agrees,  too,  with  the  mechanical  principles 
developed  in  the  Third  Paper:  That  eccentricity  of  orbit  could 
be  removed  only  by  radial  action,  and  therefore  that,  as  the 
eccentricity  decreased  and  the  radial  phenomena  became  less  and 
less,  the  difficulty  of  further  reducing  the  eccentricity  became 
greater  and  greater ;  so  that  it  is  unimaginable  that  eccentricities 
should  ever  be  reduced  to  zero,  by  activities  depending  upon 
the  eccentricity  for  their  effectiveness.  Zeros  of  pressure  and 
zeros  of  eccentricity  of  orbit  must  be  alike  dismissed  from  con- 
sideration, as  conditions  impossible  of  occurrence  in  nature,  con- 
stituting limits  which  may  be  approached  but  never  reached. 

The  same  is  true  of  infinity  of  eccentricity,  or  zero  of 
curvature,  of  orbit.  Radial  departure  between  two  bodies  can  be 
created  only  by  tangential  action  at  periastron,  as  in  the  illustra- 
tion of  the  cannon-ball.  Therefore  some  radial  component  must 
always  be  retained.  It  is  imnossible  to  imagine  tangentiallv  im- 
parted intensity  of  -adial  motion  ever  getting  to  the  point  where 
there  was  no  tangentiality ;  or  where,  in  other  words,  the  orbit 
had  ceased  to  be  a  hyperbola  and  had  become  a  straight  line. 


MECHANICAL  CONCEPTS  109 

Now,  of  the  two  sorts  of  pressure  just  mentioned,  it  is  plain 
that  outwardly  active,  or  expansive,  pressure  can  be  explained 
only  by  orbits  possessing  eccentricities  greater  than  unity.  Only 
in  such  cases  would  either  member  of  a  mass-pair  be  able  to  free 
itself  from  its  mate  or  mates,  and  fly  outwardly  until  arrested 
by  external  resistances.  Passively  resistant  pressure,  however, 
which  absorbs  energy  in  forceful  resistance  to  deformation,  but 
which  makes  no  effort  to  expand  beyond  fairly  fixed  limits,  can 
be  explained  only  in  terms  of  orbits  having  eccentricities  below 
unity;  that  is,  elliptic  orbits.  For  all  such  orbits  contain  within 
themselves  a  fixed  outward  limit  of  motion,  beyond  which  there 
will  be  no  trespass.  But  to  any  arbitrary  limitation  of  motion 
within  those  limits,  by  forces  exerted  from  without,  to  the  short- 
ening of  the  natural  length  of  the  ellipse,  there  would  be  exerted 
stout  resistance. 

Since  both  sorts  of  pressure  are  found  in  all  natural  condi- 
tions of  matter,  it  is  necessary  to  assume  that  in  all  molecular 
energy-systems  there  exist  both  elliptic  and  hyperbolic  orbits. 
Since  the  passive  form  of  pressure  is  much  greater  than  the 
expansive  sort  in  solids,  it  is  necessary  to  assume  that  in  solid 
matter  the  far  greater  portion  of  molecular  mass  is  revolving  in 
elliptic  orbit,  only  minor  fragments  following  hyperbolic  orbits. 
In  gaseous  matter,  on  the  other  hand,  it  is  necessary  to  assume 
that  the  major  portion  of  the  mass  moves  in  hyperbolic  orbit, 
only  a  minor  portion  retaining  elliptic  motion. 

It  is  only  in  the  unattainable,  limiting  case,  however,  that  all 
of  the  mass  could  assume  hyperbolic  orbits;  for  it  is  only  by 
action  which  depends  upon  mass  in  elliptic  orbit  that  any  other 
mass  may  be  given  a  superpermanent  intensity  of  energy  in 
hyperbolic  motion.  This  fact  makes  it  clear  that  no  combination 
of  natural  circumstances  could  ever  develop  in  matter  a  state 
where  all  elliptic  motion  had  ceased  and  all  the  attributes  of  a 
solid  had  disappeared.  And  this  fact,  too,  agrees  with  all  the 
observations  heretofore  drawn,  viz :  That  matter  and  energy 
depart,  in  either  direction,  from  a  central  condition  where  all 
things  are  balanced,  toward  extremes  where  one  or  the  other 
condition  becomes  exaggerated  or  suppressed,  only  with  steadily 
increasing  difficulty;  and  that  no  imaginable  conditions  or  proc- 
esses could  ever  force  things  to  the  noint  where  any  of  these 
normal  attributes  of  matter  had  become  zero.  But  such  an 


110  ENERGY 

impossible  state  of  affairs  as  matter  in  which  all  orbits  were 
hyperbolic  and  none  elliptic  would  constitute,  if  it  could  ever 
be  attained,  the  much  talked  of  "perfect  gas."  Therefore  the 
"perfect"  gas  does  not  and  cannot  exist.  Reliable  scientific 
authority  does  not  teach  that  it  does  or  could;  but  the  general 
concept  of  the  perfect  gas  has  been  used  so  irresponsibly,  and 
with  such  mischievous  results,  by  many  teachers  of  engineering 
thermodynamics,  that  we  shall  refer  again  to  its  absurdity  as  a 
concept  of  natural  matter. 

To  return  now  to  the  concept  of  expansive  pressure  as  a 
manifestation  of  hyperbolic  motion:  Such  a  concept  of  pressure 
as  a  bombardment  of  the  walls  surrounding  a  hot  substance  by  a 
multitude  of  radiating  molecules  is  by  no  means  new.  The 
trouble  with  it  is  that  it  doesn't  explain.  The  trouble  is  not  that 
such  a  bombardment  could  not  exert  the  pressure.  The  trouble 
is  that  the  pressure  is  exerted  continuously,  without  loss ;  whereas 
every  bombardment  known  to  human  experience  involves  several 
losses.  All  the  projectiles  are  lost.  All  their  energy  is  lost.  And 
usually  the  wall  itself  is  also  lost. 

It  is  of  no  use,  in  this  juncture,  to  have  it  explained  to  us  that 
the  wall  is  perfectly  elastic  and  the  projectiles  are  perfectly 
elastic,  and  that  both  wall  and  projectiles  are  indestructible.  We 
know  nothing  about  any  such  things.  The  bombardment-ex- 
planation of  pressure  has  been  to  the  author,  ever  since  he  first 
heard  it  as  a  student,  a  blind  failure  to  explain  the  obscure, 
because  attempted  with  the  aid  of  something  still  more  obscure. 
And,  so  far  as  he  can  discover,  it  has  been  equally  so  to  every 
sincere  student. 

Another  aid,  and  also  obstacle,  to  the  comprehension  of  pres- 
sure lies  in  its  similarity  to  and  its  contrast  -with  temperature. 
If  we  are  to  rely  upon  the  linear  kinetic  energy  of  the  outwardly 
flying  particles  having  hyperbolic  orbits  to  explain  pressure,  what 
is  left  to  explain  temperature?  Moreover,  temperature  has 
already  been  defined  as  this  linear  kinetic  energy.  For  pressure 
and  temperature,  while  sufficiently  alike  in  some  respects  to  be 
considered  identical,  are  yet  strongly  contrasted  in  some  other 
respects. 

Speaking  broadly,  the  active  expansive  pressure  of  the  vapors 
and  gases  is  roughly  proportional  to  temperature.  Similarly,  the 


MECHANICAL  CONCEPTS  111 

passive  resistant  pressure  of  the  solids  is  inversely  proportional 
to  temperature.  It  is  the  higher  temperatures  which  create  vapors 
and  gases,  with  their  great  expansive  and  slight  resistant  pres- 
sure. It  is  the  lower  temperatures  which  develop  solids,  exhibit- 
ing the  reverse  of  this.  In  the  more  permanent  gases  the  pro- 
portionality of  expansive  pressure  to  temperature  is  almost  exact, 
according  to  Boyle's  law,  while  the  viscosity,  or  passively  resist- 
ant pressure,  or  "solidity"  as  we  might  truly  call  it,  is  almost 
inversely  proportional.  In  the  liquids  the  vapor-tension  varies 
directly  and  widely,  though  not  exactly  proportionally,  with  the 
temperature.  In  the  solids  the  connection  between  expansive 
pressure  and  temperature  is  much  more  obscure,  but  it  is  roughly 
visible  and  it  is  nowhere  reversed. 

All  of  this  evidence,  if  it  were  not  for  the  bombardment  diffi- 
culty, would  fall  in  excellently  with  what  was  said  as  to  elliptic 
and  hyperbolic  motions.  The  almost  circular,  or  low-eccentricity, 
motion  of  the  "solid"  molecules  is  fitted  as  naturally  to  explain 
the  stout  passive  resistance  of  the  solids  as  the  highly  eccentric 
hyperbolic  motion  of  the  "gaseous"  particles  is  to  explain  active, 
expansive  pressure.  For  a  given  mass  can  most  effectively  resist 
a  deflecting  force  (without  itself  undergoing  transformation) 
when  it  is  moving  at  a  high  velocity  normally  to  that  force ;  and 
a  circle  is  normal  to  every  line  approaching  its  center  from 
without.  The  hydraulic  jets  of  the  old-fashioned  placer-mines, 
in  California,  for  instance,  were  said  to  come  from  their  nozzles 
with  such  a  velocity  that  a  man  could  not  strike  an  ax  into  the 
water.  Therefore  it  is  quite  reasonable  to  imagine  the  solid 
state  of  matter  as  consisting  of  mass-particles  situated  very 
close  together,  yet  kept  apart  by  a  very  high  velocity  of  almost 
circular  motion  about  each  other;  for  in  such  case  the  velocity 
must  increase  as  the  particles  come  more  closely  together,  or  in 
other  words,  the  hardness  and  rigidity  of  passive  resistance  to 
external  pressure  must  increase  with  the  density — which  is  just 
what  is  observed  in  nature. 

Such  a  system  would  be  elastic,  but  non-expansive.  The 
abstraction  of  energy  would  increase  both  density  and  hardness 
of  resistant  pressure,  while  the  addition  of  energy  would  soften 
and  expand  it  into  a  liquid  or  a  gas.  Everything"  about  the 
explanation  would  be  beautifully  consistent,  if  only  some  sufficient 
substitute  for  the  bombardment  could  be  found,  to  explain  the 


112  ENERGY 

method  of  application  of  these  external  forces  and  energies  to 
the  tiny  revolving  systems. 

The  key  to  this  situation,  as  also  to  the  contrast  between 
pressure  and  temperature,  lies  in  the  following  facts.  When 
pressure  performs  work,  there  is  no  transfer  of  energy  across 
the  gap  where  the  pressure  is  felt.  Whenever  and  wherever 
temperature  is  felt,  however,  there  does  occur  a  transfer  of 
energy  across  the  gap. 

The  first  of  these  statements  appears  at  first  sight  paradoxical. 
Nevertheless  it  is  true.  When  pressure  performs  work  it  is  the 
surface  bounding  the  energy  which  moves,  not  the  energy  across 
the  surface. 

To  explain,  when  pressure  performs  work  the  energy  is  most 
commonly  what  the  writer  calls  transient  energy.  In  transient 
energy  the  true  source  of  energy  is  some  more  or  less  distant 
driver,  from  which  the  energy  comes  by  some  carrier  which  is 
itself  a  mere  inert  "pusher,"  exerting  the  pressure  which  is 
under  discussion.  In  such  cases  the  amount  of  energy  trans- 
mitted is  quite  independent  of  the  mass  of  the  carrier.  Instances 
of  transient  energy  are  found  in  the  fluids  supplied  to  hydraulic 
cranes  and  pneumatic  tools,  where  a  remote  pump  or  air- 
compressor  furnishes  the  energy,  which  the  water  or  air  under 
pressure  merely  transmits  to  the  driven  piston. 

The  same  is  true  of  the  steam  of  a  steam-engine,  during 
admission.  Then  the  energy  is  furnished  solely  by  hot-water 
expansion  in  the  boiler ;  the  steam  in  the  steam-pipe  and  cylinder 
undergoes  no  energetic  change  whatever,  but  serves  merely  as  a 
fairly  frictionless  transmitter,  neither  expanding  nor  contracting, 
nor  undergoing  acceleration  or  retardation.  It  is  only  after  cut- 
off that  the  steam  itself  becomes  an  active  source  of  energy. 

It  is  transient  energy,  too,  which  is  active  in  the  water  of  a 
penstock.  It  is  only  within  the  guide-blades  and  wheel  that  the 
water's  own  energy  becomes  active.  It  is  transient  energy  which 
gives  to  the  greater  number  of  machine-parts  their  value.  All 
rods,  shafts,  chains,  belts,  etc.,  are  busy  handling  transient 
energy.  The  only  pieces  of  metal  in  which  we  utilize  the  energy 
stored  in  the  metal  itself  are  weights,  springs  and  hammer-heads. 

In  all  these  cases  it  is  obvious  that  it  is  the  surface  bounding 
the  energy,  the  surface  where  the  pressure  is  felt,  which  moves 
as  the  energy  is  transmitted,  and  not  the  energy  across  the 


MECHANICAL  CONCEPTS  113 

surface.  In  all  cases  of  transient  energy  the  energy-fund  of 
each  piece  remains  constant.  It  is  the  piece  itself  which  moves. 

In  the  case  of  non-transient  energy,  such  as  weights,  springs, 
hammers  or  bodies  of  expanding  steam  or  gas,  where  the  energy- 
fund  of  the  body  does  change,  it  is  still  true  that  no  energy  is 
transmitted  by  pressure  across  the  surface  of  contact.  It  is  only 
as  that  surface  moves  that  energy  is,  or  can  be,  transmitted. 
The  fact  that  the  energy  is  developed  within  the  body,  rather 
than  transmitted  through  it,  does  not  alter  the  case.  A  weight 
resting  at  constant  height,  a  spring  under  constant  distortion,  a 
hammer-head  which  hits  no  anvil,  or  a  body  of  steam  under 
constant  volume — none  of  them  transmit  any  energy,  although 
they  may  exert  great  pressure. 

It  is  the  essential  characteristic  of  expansive  pressure,  then, 
if  we  are  to  follow  the  mechanical  hypothesis  strictly,  that  it 
consists  of  a  bombardment — because  it  is  associated  only  with 
the  hyperbolic  paths  of  particles  which  would  not  turn  back 
except  for  the  external  pressure — but  of  a  bombardment  in 
which  no  energy  is  transferred. 

Now  such  a  bombardment  is  very  different  from  anything 
known  in  military  experience.  Such  a  bombardment  would 
require  the  party  bombarded  to  kindly  catch  all  projectiles,  with- 
out impact  or  friction,  and  to  return  them  with  velocity  con- 
served and  direction  of  motion  reversed.  Such  an  action  has 
been  provided,  in  the  explanations  of  pressure  commonly  given, 
by  the  assumption  of  "perfect  elasticity."  But  the  reversal  of 
direction  of  motion  with  perfect  conservation  of  energy  is  only 
found  in  nature,  it  has  repeatedly  been  pointed  out,  in  the 
mutual  revolution  of  mass-portions  around  one  another.  There- 
fore, if  the  hypothesis  that  heat  is  a  mode  of  motion  is  to 
remain  intact,  pressure  must  be  regarded  as  the  result  of  the 
radiation  from  the  surface  in  question  of  a  continuous  flow  of 
tiny  projectiles ;  but  these  projectiles,  instead  of  actually  striking 
and  rebounding  from  contact  with  the  particles  of  the  other 
body  (for  pressure  can  be  felt  only  between  two  bodies),  must 
be  imagined  as  being  met  half-way  by  a  similar  swarm  of  pro- 
jectiles radiated  from  the  other  body,  as  revolving  about  them 
in  a  "swing-opposite-partners"  fashion,  returning  home  with 
pressure  exchanged  but  energy  conserved. 

According  to  this,  there  can  never  occur  what  is  commonly 


114  ENERGY 

called  "contact"  between  mass-portions.  All  action  is  at  a  dis- 
tance. Contact,  so-called,  is  merely  the  approach  of  two  systems 
of  particles  into  such  propinquity  that  they  begin  to  feel — or 
rather,  to  manifest  perceptibly  to  our  crude  senses — the  repulsive 
effect  of  the  bombardment  of  projectiles  from  the  other  body. 
In  gases  and  vapors  these  projectiles  travel  far  and  embody 
considerable  mass;  most  of  the  mass  of  the  body  is  in  projectile 
form;  therefore  the  repulsive  effect  is  far  reaching.  In  solids, 
on  the  other  hand,  the  projectiles  are  few  and  of  short  range. 
The  major  portion  of  the  mass  of  each  molecule  is  engaged  in 
elliptic  motion,  exerting  no  expansive  pressure.  Such  a  central 
and  self-contained  portion  of  the  solid  molecule  is  called  the 
nucleus,  in  contrast  with  the  projectiles  which  it  sends  out  upon 
hyperbolic  orbits.  In  the  case  of  solids  the  expansive  pressure 
of  these  projectiles  can  scarcely  be  felt.  But  as  soon  as  the 
external  pressure  drives  these  feeble  projectiles  home  upon  the 
rapidly  revolving  (and  therefore  rigid)  nucleus,  the  resistance  of 
the  latter  promptly  becomes  considerable. 

An  excellent  simile  for  this  situation  is  the  exchange  of 
pressure  between  bodies  of  troops  in  war.  There  the  sense  of 
contact  is  exchanged  by  means  of  flying  projectiles,  or  bullets. 
Each  body  experiences  a  force  of  repulsion  from  the  other  by 
means  of  these  projectiles,  long  before  the  two  bodies  come 
visibly  into  contact.  An  observer  of  military  action  from  some 
commanding  height,  who  was  familiar  with  primitive  warfare 
but  knew  nothing  of  guns  and  bullets,  would  be  much  surprised 
and  puzzled  in  seeing  two  orderly  masses  of  men,  moving  in 
opposing  directions  with  antagonistic  aims,  slow  down  as  they 
approached  within  a  considerable  distance  of  each  other,  hesitate, 
waver,  desist  from  their  predetermined  purpose,  break  into  small 
and  irregular  detachments  at  the  points  of  closest  approach,  take 
on  lateral  motions  of  various  sorts,  and  finally  the  smaller  body 
reverse  its  direction  of  motion  in  response  to  the  repulsion  of  the 
larger,  and  depart  with  a  velocity  higher  than  that  of  approach — 
all  of  which  is  quite  similar  to  what  we  observe  in  the  "contact" 
between  solid  portions  of  matter.  The  imaginary  observer  of 
all  this  could  not  see  the  flying  bullets  which  were  the  active 
agents  in  this  transfer  of  pressure  from  army  to  army,  and  he 
would  be  entirely  at  a  loss  to  explain  it,  no  matter  how  wide 
had  been  his  experience  with  warfare  conducted  with  swords, 


MECHANICAL  CONCEPTS  115 

spears    and    chariots — until    some    one    mentioned    the    bullets. 

Now  the  impulse  carried  by  the  bullets  is  a  moral  one,  rather 
than  physical,  and  is  dependent  as  much  upon  the  accuracy  and 
timeliness  of  their  discharge  as  upon  their  mass  and  velocity. 
So  this  simile  is  truly  a  simile,  rather  than  an  accurate  scientific 
analogy.  Yet  it  may  be  extended  usefully  to  the  contrasts 
between  gases  and  solids.  An  army  progressing  freely  through 
territory  where  opposition  is  not  imminent,  divided  into  many 
separate  bodies,  with  extended  columns,  flying  detachments  of 
cavalry,  scouting  and  foraging  parties,  skirmish-lines,  etc.,  may 
be  likened  to  a  gas.  It  will  occupy  all  the  space  it  can. 

Such  an  army,  meeting  another  such,  would  feel  the  pressure 
of  contact  very  gradually.  A  skirmish-line  would  be  driven 
back  here  or  there;  scouting  parties  and  flying  detachments 
would  become  more  careful  and  stay  nearer  home.  But  only 
after  appreciable  time  would  these  outlying  features  be  driven 
into  consolidation  with  the  main  body,  and  the  entire  mass  be 
compressed  into  a  form  so  dense  that  pressure  felt  at  any  one 
point  would  be  transmitted  with  promptness  throughout  the  entire 
mass. 

The  armies  thus  condensed,  as  in  active  conflict  in  difficult 
country,  where  the  enemy's  pulse  is  not  easily  felt,  might  be 
likened  to  solids.  In  such  case  there  would  be  little  premonition 
of  contact,  until  it  occurred  with  considerable  force.  Then  there 
would  be  little  elasticity.  Instead  of  repulsion  in  visible  reverse- 
motion  or  diversion  of  path,  the  exchange  of  energy  would  be 
sharp  and  sudden,  and  would  occur  with  great  loss.  Contact 
would  become  synonymous  with  impact.  The  balance  of  power 
would  be  established  through  grinding  wear,  with  the  partial 
destruction  of  the  reacting  bodies,  in  disintegration  and  heat  of 
conflict,  rather  than  with  a  graceful  yielding  to  the  pressure  from 
without  and  a  gradual  accumulation  of  available  mass  against 
the  point  of  contact,  as  is  the  case  with  elastic  bodies. 

In  all  these  ways  the  likeness  between  the  contact  of  bodies 
of  troops  and  that  between  bodies  of  physical  mass  in  motion  is 
very  great.  In  both  cases  what  is  called  contact  proves,  upon 
careful  examination,  to  be  no  contact  at  all.  It  is  merely  an 
approach  into  sufficient  propinquity  so  that  a  perceptible  force 
of  repulsion  is  experienced.  And  wherever  the  word  perceptible 
is  used  it  refers  as  much  to  the  thing  doing  the  perceiving  as  it 


116  ENERGY 

does  to  the  thing  perceived.  There  is  no  end  of  things  in  nature 
which  are  imperceptible  to  our  fairly  crude  human  senses,  when 
applied  directly,  which  have  been  made  plain  either  by  the  greater 
sensitiveness  of  inanimate  instruments  or  by  scientific  analysis. 
And  apparently  the  universality  of  the  manifestation  of  repulsive 
pressure  throughout  the  universe,  between  each  two  bodies,  is 
one  of  these. 

The  further  development  of  this  idea  of  the  transmission  of 
pressure  by  projectiles,  as  the  only  possible  explanation  of  ther- 
mal phenomena  which  is  in  accord  with  Newtonian  mechanics, 
will  be  deferred  for  a  continuation  of  this  chapter. 


CHAPTER  X. 
THE  MECHANICAL  CONCEPT  OF  PRESSURE  (Continued). 

The  mental  picture  which  must  be  formed  of  the  average 
molecule,  in  order  to  prosecute  the  mechanical  analysis  of  heat, 
is  as  complex  and  varied  as  the  aspect  of  an  army-corps  in 
active  campaign — to  complete  the  simile  which  was  introduced 
in  the  preceding  paper.  It  must  itself  consist  of  many,  very 
many  separate  portions. 

The  major  portion  of  this  mass,  in  the  solids  and  liquids,  at 
any  rate,  will  be  devoted  to  the  formation  of  a  central  body 
which  may  be  called  the  nucleus.  This  nucleus  is  itself  complex 
in  structure,  formed  of  many  parts,  which  may  separate,  upon 
occasion,  in  independent  action.  Its  internal  motions  are  more 
or  less  mobile  and  fluid.  Yet  it  embodies  a  greater  degree  of 
consistency,  and  approaches  more  nearly  to  the  condition  of  a 
solid,  than  the  more  out-lying  portions.  In  the  mechanical  con- 
cept of  heat  this  nucleus  must  be  conceived  as  composed  of 
particles  revolving  in  elliptic  orbits,  many  of  them  of  an  eccen- 
tricity approaching  zero.  In  this  way  the  nucleus  may  be 
understood  to  possess  a  unified  existence  of  its  own,  of  fairly 
permanent  dimensions,  until  its  internal  equilibrium  is  upset 
and  its  unity  broken  up,  by  its  invasion  with  sufficient  force  by 
some  disturbing  mass  from  without. 

Surrounding  this  nucleus  is  a  swarm  of  particles  possessing 
hyperbolic  orbits.  These,  to  distinguish  their  activities  from  that 
of  the  nucleus,  will  be  called  the  satellites.  Nevertheless,  it  is 
not  necessary  to  assign  any  other  distinguishing  feature  to  the 
satellites  than  their  high  eccentricity  of  orbit.  They  may  be 
regarded  as  capable  of  becoming  absorbed  at  any  time  by  the 
nucleus,  or  of  being  formed  from  nuclear  particles.  Indeed,  it 
is  natural  to  assume  that  this  process  of  transfer  of  particles, 
from  nucleus  to  satellitic  swarm  and  the  reverse,  as  one  particle 
or  another  gains  or  loses  the  intensity  of  energy  requisite,  is 
going  on  at  all  times.  This  may  occur  as  readily  as,  in  the 

117 


118  ENERGY 

similar  army-corps,  men  are  constantly  being  detached  from 
head-quarters  for  assignment  to  distant  posts  of  duty;  while 
other  men,  these  duties  done,  are  as  continually  returning,  for 
re-absorption  by  the  central  body. 

Nor  is  it  necessary  to  imagine  any  sharp  definition  between 
the  nuclear  swarm  and  the  satellitic  swarm.  The  nucleus,  on  the 
one  hand,  may  possess  some  members  which,  although  moving  in 
almost  circular  orbits,  yet  lie  far  out  away  from  the  main  mass. 
Such  instances  occur  in  our  own  solar  system,  for  instance,  in 
such  planets  as  Neptune  and  Uranus,  which  have  not  completed 
one  of  their  "years"  since  the  dawn  of  modern  astronomy;  or 
in  one  of  the  moons  of  Jupiter,  which  is  so  far  distant  from 
that  planet  that  it  is  barely  caught  permanently,  against  the  sun's 
attraction. 

The  satellitic  swarm,  on  the  other  hand,  may  possess  many 
members  whose  great  eccentricity  of  orbit  carries  them  as  near 
to  the  center  of  mass  at  periastron  as  they  are  distant  at  apastron. 
Such  particles,  which  would  correspond  to  the  comets  of  our 
solar  system,  would  alternately  dive  into  the  very  center  of  the 
nuclear  swarm,  and  then  depart  as  remotely  into  space  as 
external  mass-systems  might  permit.  The  instance,  in  our  solar 
system,  of  the  Great  Comet  of  1882  has  already  been  mentioned. 
Here  is  a  vast  mass  which  last  experienced  its  maximum  separa- 
tion from  the  mass-center  of  the  solar  system  when  Columbus 
was  struggling  into  manhood.  At  that  time  the  comet  was  at 
aphelion,  far  beyond  the  orbit  of  the  most  remote  known  planet, 
drifting  lazily  in  parallel  with  the  planetary  orbits.  But  whereas 
the  planets  possess  sufficient  tangential  motion  to  prevent  them 
from  falling  toward  the  sun,  this  comet  did  not.  It  began  to  fall 
before  Columbus  set  sail  for  the  Indies,  and  for  four  hundred 
years  thereafter  it  fell  continually,  with  velocity  increasing  each 
second,  through  unknown  millions  of  miles,  toward  the  sun. 
Finally,  in  September,  1882,  it  reached  its  mean  energetic  con- 
dition, very  near  the  sun.  Within  ninety  minutes  thereafter  it 
had  further  transformed  into  kinetic  form  an  equal  amount  of 
energy  with  that  transformed  during  the  preceding  four  cen- 
turies, and  had  reached  its  extreme  energetic  condition,  at  peri- 
helion. Within  a  second  period  of  ninety  minutes  this  gigantic 
scale  of  energy-transformation  had  been  reversed,  the  second 
mean  energetic  condition  had  been  passed,  and  the  comet  was 


MECHANICAL  CONCEPTS  H9 

away  upon  another  eight  centuries  of  outward  and  return  motion, 
almost  purely  radial  in  character. 

Yet  a  molecular  particle  acting  similarly  to  this  must  be 
classed  as  belonging  to  the  "solid"  nucleus,  rather  than  to  the 
expansive-pressure-exerting  satellitic  swarm,  because  its  orbit  is 
elliptic.  The  influence  of  no  external  mass-system  is  needed  to 
return  this  satellite  to  the  sun.  No  external  mass-system  would 
feel  its  pressure  until  it  had  encroached  upon  definite  dimensions 
of  our  solar  system. 

Therefore  the  molecular  mechanical  system  is  to  be  imagined 
as  consisting  of  all  sizes  and  velocities  of  mass-portion,  moving 
in  all  sorts  of  orbits.  Some  will  move  in  almost  circular  orbits 
of  small  radius  and  very  high  velocity,  others  in  the  same  form 
of  orbit  with  large  radii  and  low  velocities.  Some  will  move 
in  ellipses  of  high  eccentricity,  with  large  and  small  mean  ener- 
getic distances.  Some  will  move  in  hyperbolic  orbits,  tending  to 
lose  themselves  from  the  system  except  as  they  are  returned  to 
it  by  outside  forces.  Indeed,  the  mechanical  idea  of  the  molecule 
must  also  include  the  constant  loss  of  some  of  these  smaller 
particles  from  the  molecule,  their  mass  being  made  up,  under 
average  conditions,  by  an  equal  absorption  of  freely  roving  par- 
ticles lost  by  other  molecular  nuclei — just  as  an  army-corps  both 
loses  and  gains  men  continually,  in  the  form  of  veterans  disabled 
home  and  raw  recruits  received,  and  in  the  form  of  prisoners  lost, 
taken  and  exchanged  with  the  enemy. 

In  a  general  way,  the  proportion  of  satellitic  energy  to  that 
embodied  in  the  elliptic  orbits  of  the  nucleus  is  given  by  the 
proportion  of  "external  work"  visible  in  the  substance  in  question. 
Thus,  in  the  case  of  steam,  a  glance  at  the  steam-tables  will 
show  that  the  proportion  of  any  increment  of  heat  going  respect- 
ively to  internal  and  external  work  differs  widely  in  ice,  water, 
superheated  steam  and  the  mixture  of  dissociated  hydrogen  and 
oxygen.  If  we  remember  that  the  "external"  work  is  that  por- 
tion of  the  energy  which  is  devoted  to  holding  at  bay  the  sur- 
rounding bodies  which  press  in  upon  the  water,  steam,  etc.,  it 
is  plain  that  the  work  involved  is  that  stored  in  the  satellitic 
swarm.  During  the  flight  of  each  satellite  its  fund  of  energy  is 
stored  in  kinetic  form.  During  the  "swing-opposite-partners" 
reversal  of  motion  at  the  far  end  of  the  route,  where  the  pro- 
'jectile  comes  into  "contact"  with  the  external  mass,  such  as  a 


120  ENERGY 

boiler-shell  or  engine-piston,  which  is  being  pressed  back  by  the 
hot,  elastic  body,  this  kinetic  energy  takes  the  form  of  '  'pro- 
pinquity," or  intensity  of  spacial  approach.  Immediately  there- 
after the  motion  is  reversed,  toward  the  original  nucleus,  and 
the  energy  becomes  kinetic  again. 

Therefore,  the  prominence  of  external  work  in  the  energetic 
action  of  any  body  is  an  excellent  criterion  of  the  proportion  of 
the  mass  of  each  of  its  molecules  which  is  engaged  in  satellitic, 
or  hyperbolic,  orbital  motion,  as  contrasted  with  that  which  is 
involved  in  nuclear,  or  elliptic,  orbital  motion. 

In  general,  both  the  nucleus  and  the  satellites  will  tend  to 
arrange  themselves  in  a  generally  spherical  form.  While,  no 
doubt,  an  individual  variation  of  form  away  from  the  spherical 
could  be  detected  in  each  molecule,  could  it  be  examined  closely, 
yet  the  general  tendency  of  all  particles  would  be  to  dispose 
their  departures  from  the  center  equally  in  all  directions,  and  to 
about  equal  distances. 

So  soon  as  two  molecules  should  approach  within  "per- 
ceptible" propinquity,  the  form  of  each  of  their  internal  orbits, 
and  the  general  form  of  each  molecule,  would  be  altered,  by  the 
attraction  of  the  mass  of  the  other  molecule.  The  situation  may 
be  illustrated  by  Fig.  9,  wherein  the  several  typical  forms  of 
orbits,  and  their  perturbation  by  the  mutual  approach  of  the  two 
molecular  nuclei,  A  and  B,  is  shown  by  dotted  orbits  marked 
a,  b,  c  and  d. 

Many  of  the  orbits,  including  most  of  those  embodied  in  the 
nucleus,  which  before  were  almost  circular,  would  experience,  as 
the  result  of  this  attraction,  merely  a  lengthening  of  their  major 
axes  toward  the  other  molecule.  These  are  shown  at  a  and  b. 
Other  orbits,  such  as  those  shown  intersecting  at  c,  which  before 
had  been  ellipses  much  elongated  toward  the  other  molecule, 
would  find  their  major  axes  shortened  at  one  end,  the  end  nearer 
the  other  molecule,  by  the  mutual  attraction  between  the  particles 
themselves  and  their  mutual  revolution  about  each  other  at  c,  in  a 
little  periastron  of  their  own.  Indeed,  in  the  greater  number  of 
such  cases  the  orbits  would  before  have  been  the  hyperbolic  ones 
of  satellites  exerting  expansive  pressure  outwardly ;  but  now  they 
are  drawn  down  into  ellipticity  by  their  own  mutual  attraction. 
Such  orbits,  however,  would  not  be  geometrically  perfect  ellipses. 


MECHANICAL  CONCEPTS  121 

Other  orbits,  such  as  d,  are  those  which,  originally  hyperbolic 
in  reference  to  A  or  B  alone,  are  now  drawn  down  into  ellip- 
ticity  by  the  combined  attraction  of  both,  but  perform  alternate 
periastrons  about  first  one  and  then  the  other.  Nor  are  these 
orbits  necessarily  elliptic.  They  may  be  of  the  form  of  a  figure 
eight,  or  looped  in  still  more  intricate  form,  when  many  more 
than  two  interacting  molecular  nuclei  are  involved.  The  entire 
process  of  interchange  of  pressure,  indeed,  may  be  likened  to 


FIG.  9. 


some  of  the  figures  of  the  old-fashioned  quadrille,  in  which  "right 
and  left"  or  'ladies'  chain"  or  "dos  a  dos"  furnish  opportunity 
for  sturdy  nuclear  gentlemen  to  set  delicate  satellitic  ladies  flit- 
ting about  orbits  of  the  most  intricate  and  confusing  character. 
In  which  connection  it  is  of  further  interest  to  note  that  all  of 
these  old  dance-forms,  like  most  savage  dance-figures,  arose  from 
the  primitive  custom  of  amusement  with  mimic  warfare,  as  in 
the  joust  or  tournament,  in  which  advance  and  retreat,  skirmish 
and  charge  en  bloc,  were  so  arranged  as  to  exchange  pressure, 
but  maintain  equilibrium,  between  two  opposing  parties. 

Any  one  of  these  satellites,  at  every  point  in  its  orbit,  is 
subject  to  two  attractive  forces,  one  drawing  it  toward  the 
nucleus  A  and  the  other  toward  B.  In  each  case  the  attraction 
reacts  upon  the  nucleus  with  the  same  force  that  it  exerts  upon 
the  satellite.  The  two  attractive  forces,  however,  are  seldom 


122  ENERGY 

equal.  At  any  such  point  as  e,  for  instance,  the  two  forces 
would  bear  the  ratio 

$-  <s* 

assuming  that  the  masses  of  A  and  B  are  equal.  But  this  same 
satellite,  if  its  orbit  were  of  the  d-class  (or  some  chance  equiva- 
lent in  the  horde  of  satellites  if  it  were  not),  would  also  be 
found  later  at  the  point  f,  engaged  in  periastron,  around  the 
other  nucleus  B.  The  point  /  being  symmetrically  opposite  to  e, 
the  forces  developed  at  /  will  be  equal  and  opposite  to  those  at  e. 
Only  now  the  larger  force  is  exerted  upon  the  nucleus  B  instead 
of  upon  A. 

The  resultants  of  all  of  these  different  forces  acting  upon  the 
two  nuclei  A  and  B  will  always  tend  to  alinement  with  the  axis 
AB  joining  the  two  bodies,  and  will  always  be  separative,  or 
repulsive,  in  its  direction.  The  more  nearly  the  two  bodies 
approach,  the  greater  will  be  the  proportion  of  the  mass  of  each 
which  engages  in  the  sort  of  modified  orbit  which  develops  these 
repulsive  forces. 

If  the  two  systems,  each  consisting  of  nucleus  and  swarm  of 
satellites,  should  be  generally  circular  or  spherical  in  outline 
(according  to  whether  confined  or  not  to  a  single  plane),  as  is 
our  solar  system,  and  should  be  of  uniform  density,  these 
repulsive  forces  would  vary  inversely  as  the  square  of  the 
distance  between  the  nuclei.  While  there  is  no  need  for  forcing 
any  particular  assumption  as  to  the  form  of  the  molecule,  this 
much  is  said  to  show  that  any  ordinary  laws  of  variation  of 
force  with  propinquity  may  be  explained  in  a  sensible  fashion,  in 
terms  of  such  a  mechanical  system.  In  general,  it  may  be  said 
that  any  two  such  systems,  upon  approach,  would  first  be  elon- 
gated in  the  line  of  their  approach,  which  would  increase  their 
radial  susceptibility,  or  elasticity,  in  that  direction,  and  would 
afterward  be  mutually  repelled,  in  a  perfectly  elastic  manner,  by 
forces  increasing  with  some  power  of  the  propinquity. 

Early  in  this  series  of  papers  it  was  shown  that  true  elas- 
ticity can  be  conceived,  in  consistency  with  the  Newtonian 
mechanics,  only  when  the  relative  motion  of  a  mass-pair  is 
reversed  by  the  circumrevolution  of  the  two  members.  It  now 
becomes  clear  how  perfect  elasticity  may  be  manifested  between 
two  complex  mass-systems  by  the  circumrevolution  of  only  a 


MECHANICAL  CONCEPTS  123 

portion  of  each  mass-system  about  a  similar  portion  of  the 
other.  The  proportion  of  the  total  mass  which  is  thus  neces- 
sarily involved  in  penetration  into  and  circumrevolution  about 
the  other  depends,  of  course,  only  upon  the  intensity  of  the 
relative  motion  between  the  systems;  that  is  to  say,  upon  the 
amount  of  V2  in  proportion  to  the  M±  +  M2  present.  It  is  hence 
easy  to  infer  how  it  is  that  intensity  of  energy  is  a  controlling 
factor  in  determining  energy-transformation;  for  the  penetra- 
tion of  more  than  a  certain  proportion  of  each  system  into  the 
other  would  break  up  the  internal  equilibrium  of  both,  resulting 
in  either  their  amalgamation  or  their  permanent  rupture ;  in  short, 
in  their  "transformation." 

In  imagining  any  such  systems  as  those  of  Fig.  9  as  con- 
stituting actual  molecules,  it  is  to  be  remembered  that  the  pro- 
portion of  satellites  to  nucleus  may  be  widely  variable.  In  the 
case  of  solids  the  nuclei  must  embody  the  greater  portion  of  the 
mass  of  the  molecule.  They  must  be  small  in  diameter,  very 
dense,  and  very  rigid  from  their  high  speed  of  rotation.  The 
satellites  to  correspond  would  be  few,  but  would  possess  great 
density  and  high  velocity.  In  the  case  of  gases,  the  nuclei  would 
possess  large  diameters,  low  densities  and  relatively  low  periph- 
eral speeds.  The  satellites  would  be  relatively  of  much  greater 
mass  than  in  solids. 

If  the  proportion  of  satellitic  to  total  mass  of  the  molecule 
be  represented  by  k,  then  the  repulsive  force  developed  by  two 
approaching  systems  would  vary,  not  only  inversely  with  some 
power  of  the  distance,  but  proportionally  to  M2(k  —  k2).  This 
expression  reaches  its  maximum  when  k=J  or  when  the  mass 
is  equally  divided  between  nucleus  and  satellites.  It  is  therefore 
to  be  expected  that  the  possibilities  for  elastic  force  of  repulsion 
should  be  the  greatest  in  some  intermediate  stage,  such  as  the 
liquid,  where  the  satellites  would  bear  a  medium  proportion  to 
nucleus,  rather  than  in  the  extreme  conditions  of  solid  or  gas, 
where  either  nuclear  or  satellitic  mass  preponderates.  This 
seems  to  be  true  in  nature.  In  the  gases  the  elasticity  is  well- 
nigh  perfect,  but  the  force  of  repulsion  is  small.  In  the  solids 
the  force  of  repulsion  is  very  great,  but  the  elasticity  is  quite 
imperfect.  In  the  liquids,  however,  the  force  resisting  com- 
pression is  very  great,  and  at  the  same  time  the  elasticity  is 
excellent. 


124  ENERGY 

Should  any  pair  of  mass-systems  such  as  Fig.  9  be  forced,  by 
great  energy  and  directness  of  impact,  into  closer  propinquity 
than  would  permit  the  nuclei  to  remain  intact,  they  will  become 
broken  up.  They  may  then  separate  in  fragments,  or  coalesce 
into  a  new  nucleus,  of  larger  dimensions  and  of  the  same  or 
different  form  of  arrangement.  It  is  thus  that  the  processes  of 
welding  solids,  mixing  liquids  and  gases,  and  impact  and  friction 
can  be  explained.  Chemical  combination  may  also  be  imagined 
as  explained  in  the  same  way,  although  the  premises  must  then 
be  stated  in  a  much  more  intricate  manner.  Obversely  the 
accumulation  within  any  one  molecule  of  too  much  intensity  of 
energy  might  easily  lead  to  its  bursting  or  splitting  into  two  or 
more  separate  molecules,  of  the  same  or  of  different  chemical 
nature.* 

Another  result  imaginable  from  the  too  close  collision  of 
nucleus  with  nucleus  would  be  the  production  of  more  satellites, 
from  the  fragments  of  the  ruptured  nuclei.  The  importance  of 
this  natural  mechanical  action,  as  an  explanation  of  heat-forma- 
tion by  impact,  was  pointed  out  in  the  Seventh  Paper,  on  "What 
is  Heat?"  It  is  now  to  be  pointed  out,  in  addition,  that  this 
process,  like  all  other  true  energetic  processes,  develops  a  con- 
dition of  stable  equilibrium.  That  is  to  say,  the  impact  of  the 
nuclei  was  due  to  a  paucity  of  satellites.  The  result  of  the 
collision  is  to  reduce  the  nuclei  partially  to  satellites,  thus  making 
good  the  deficit  of  satellites  and  diminishing  the  chances  of 
another  similar  collision's  occurring. 

The  beautifully  stable  balance  thus  established  pervades  all 
nature.  It  has  already  been  referred  to  briefly,  in  our  primary 
definition  of  heat.  Its  wide  results  will  be  taken  up  later  at 
length,  in  the  chapter  upon  Thermal  Equilibrium  (Chapter  XV). 

In  discussing  the  action  of  these  colliding  systems  it  is  hardly 
necessary  to  point  out  that  the  distance-factor,  for  instance,  must 
possess  a  vastly  different  scale  in  mechanical,  celestial  and  thermal 
energetics,  respectively.  Distances  which  constitute  mean  ener- 
getic ones  here  on  the  earth's  surface,  in  the  applied  mechanics 
of  engineering,  would  constitute  prodigious  extremes  of  separa- 
tion when  measured  between  molecules,  but  corresponding 

*The  mechanics  of  this  hypothesis  has  been  beautifully  developed  in 
the  theory  of  the  evolution  of  stellar  nebulae  into  dumb-bell  form,  with 
their  ultimate  consolidation  into  "double"  stars. 


MECHANICAL  CONCEPTS  125 

extremes  of  propinquity,  or  concentration  of  matter,  when  meas- 
ured between  planets. 

It  is  necessary  to  point  out,  however,  that  a  similar  range  in 
the  mean  energetic  values  of  the  other  factors  of  energy,  such  as 
force,  velocity,  density,  etc.,  is  to  be  expected  as  one  passes  from 
mechanical  to  celestial  energetics,  on  the  one  hand,  or  to  molec- 
ular energetics  on  the  other.  Yet  there  is  no  reason  to  suppose, 
from  these  differences  in  degree,  that  the  principle  of  action  is 
at  all  different.  For  instance,  there  has  been  every  reason  to 
suppose  that  the  principles  of  mechanics  were  applicable  to  the 
explanation  of  thermal  phenomena.  And  yet,  if  so,  how  are  the 
enormous  forces  which  are  needed  to  account  for  thermal  ener- 
gies to  be  explained?  Thus,  one  pound  of  carbon,  in  the  form 
of  coal,  by  virtue  of  its  separation  from  2f  pounds  of  oxygen 
in  the  atmosphere,  embodies  in  the  total  mass  of  3!  pounds 
some  14,600  B.t.u.  or  11,360,000  foot-pounds  of  potential  energy. 
All  of  this  is  released  kinetically  upon  letting  the  carbon  and 
oxygen  fall  together,  or  "burn."  Equating  this  quantity  to  the 
equation  for  potential  energy,  in  which  the  quantity-factor  MXM2 

would   equal    — - —  x  — ~  0.002 s8,   it  results   that   S0,  the 

32.16       32.16 

closest  distance  of  approach  between  carbon  and  oxygen,  must 
be  only  about  io~l6  inches.  But  this  is  much  closer  than  the  mean 
diameter  of  molecules,  as  estimated  from  other  sources,  has 
been  accepted  as  being.  According  to  Professor  J.  J.  Thomson, 
the  diameter  of  the  molecule  is  about  4  X  i°~9  inches,  while  that 
of  the  electron  is  about  4  X  io"14  inches.* 

From  this  the  conclusion  is  driven  inevitably  to  one  of  two 
things.  First,  the  carbon  and  oxygen  must  each  be  subdivided 
into  particles  much  smaller  than  a  single  molecule,  which  so  pene- 
trate the  other's  molecular  swarms  as  to  attain  a  mean  pro- 
pinquity far  greater  than  that  permitted  by  molecular  diam- 
eters. Secondly,  the  density  of  mass  in  the  molecular  particles 
is  far  greater  than  that  of  matter  as  we  are  familiar  with  it  in 
the  solar  system.  Probably  both  statements  are  true. 

This  is  the  equivalent  of  saying  that  what  has  hitherto  been 
considered  and  measured  in  diameter  as  a  molecule  is  not  to  be 
considered  as  a  solid  homogeneous  sphere,  but  as  a  multiplex 

'Engineering  (London),  March  19,  1909,  page  391. 


126  ENERGY 

mass-system,  containing  much  space  as  well  as  material  particles. 
It  was  Professor  Rowland,  we  believe,  who  said,  many  years 
ago,  that  "a  grand  piano,  in  comparison  with  a  molecule,  is  sim- 
plicity itself."  As  to  what  may  be  the  proportions  between 
space  and  matter  in  the  molecule,  nothing  may  be  said  with 
confidence.  Pursuing,  however,  the  plan  hitherto  followed,  of 
deducing  all  our  molecular  ideas  from  celestial  mechanics,  the 
molecule  should  follow  the  proportions  of  our  solar  system. 

But  in  speaking  of  densities,  in  connection  with  solar  systems 
and  molecules,  it  is  too  commonly  assumed  that  the  densities  of 
individual  planets,  such  as  our  earth,  constitute  our  sole  and 
proper  guide  to  the  densities  of  molecular  matter.  But  if  mole- 
cules are  to  be  likened  to  solar  systems,  a  collection  of  molecules, 
such  as  ordinary  matter,  must  be  likened  to  a  collection  of  solar 
systems.  The  density  of  the  particles  composing  the  molecule 
must  bear  the  same  proportion  to  the  density  of  the  entire  mole- 
cule, or  to  a  collection  of  molecules,  as  the  density  of  planetary 
mass  bears  to  the  mean  density  of  a  solar  system  or  a  galaxy. 

What,  then,  is  the  mean  density  of  our  solar  system  ?  Assum- 
ing that  the  orbit  of  the  most  distant  planet,  Neptune,  constitutes 
the  outer  limit  of  the  solar  system — although  many  comets  con- 
trolled by  our  sun  far  exceed  these  limits — and  remembering 
that  a  molecule  is  a  system  occupying  space  of  three  dimensions, 
whereas  the  orbits  of  the  planets  occupy  virtually  only  space  of 
two  dimensions — the  mean  density  of  the  space  occupied  by  the 
solar  system  is  easily  computed  as  being  less  than  that  of  our 
solid  earth  by  about  the  ratio  io12.  This  means  that  the  mean 
density  of  such  a  system,  when  perceived  from  far  enough 
without  so  that  it  appears  as  a  unit,  is  about  that  of  one  three- 
hundred-millionth  of  an  atmosphere,  or  far  rarer  than  the  matter 
within  an  electric-light  bulb,  or  even  a  Crookes  tube.  If  these 
same  proportions  are  to  apply  to  a  molecule  of  carbon  dioxid,  for 
instance,  then  the  mean  density  of  the  smallest  particles  which 
engage  in  heat-motion,  and  which  will  retain  the  characteristics 
of  carbon  and  oxygen,  respectively,  must  be  some  million  millions 
of  times  greater  than  that  of  our  solid  earth. 

There  is  no  intention  here  to  place  any  weight  upon  the  exact 
arithmetical  aspect  of  these  proportions.  All  that  is  intended  is 
that,  if  our  ideas  of  heat-energy  are  to  be  explained  in  terms  of 
mechanics,  and  if  our  mechanics  are  to  be  drawn  from  the  only 


MECHANICAL  CONCEPTS  127 

known  source  of  true  mechanics — celestial  mechanics — then  there 
appears  to  be  no  difficulty  in  explaining  the  heat-energy  stored 
chemically  in  the  dissociation  of  elements,  as  due  to  simple 
gravitation  between  mass-particles.  It  is  just  as  probable  that 
the  density  of  such  related  particles  is  many  millions  of  times 
greater  than  the  densities  of  matter  familiar  to  human  discern- 
ment as  it  is  that  molecular  dimensions  and  distances  should  be 
many  millions  of  times  smaller.  It  is  the  wide  range  in  densities 
thus  opened  before  us,  as  we  deal  with  more  and  more  finely 
subdivided  mass,  which  makes  it  possible  to  explain  even  the 
energy  of  radium,  which  embodies  a  million  times  the  energy  of 
an  equal  mass  of  oxygen  and  hydrogen,  in  terms  of  mechanical 
energetics.  Similarly,  the  explanation  of  the  heat-energy  stored 
in  the  physical  disgregation  of  matter  in  vaporization  or  fusion 
is  still  easier,  for  the  amount  of  energy  stored  per  unit  of  aggre- 
gate mass  is  much  less.  If  it  should  happen  that  not  all  of  the 
vast  range  of  dimensions  which  the  above  figures  show  to  be 
open  to  the  imagination  is  necessary  for  the  explanation,  so 
much  the  better.  If  it  should  happen,  too,  that  the  dimensions, 
densities,  etc.,  which  are  requisite  for  the  mechanical  explanation 
of  heat  do  not  immediately  fit  the  estimates  which  have  been 
drawn  from  other  considerations,  the  discrepancy  is  for  the 
physicists  to  explain.  The  doctrines  of  the  Newtonian  mechanics, 
and  of  heat  as  a  mode  of  motion  and  separation  within  mass,  are 
both  of  them  now  too  broadly  founded  in  scientific  experience 
for  them  to  bend  easily  to  meet  other  empiricisms  which  may  be 
discrepant  therewith.* 

A  second  intention  of  these  statements  of  enormous  ratios  is 
to  impress  upon  the  mind  the  fact  that  the  solid  forms  of  matter 
with  which  the  mechanics  of  engineering  is  chiefly  concerned 

*The  writer  wishes  to  repeat  and  emphasize  here  the  caution  which 
appears  in  the  preface,  that  these  statements  are  not  to  be  regarded  by  the 
student  as  coming  from  one  who  has  investigated  molecular  activities  at 
length  in  the  physical  laboratory.  They  are  founded  merely  upon  the 
premises  which  have  been  frequently  repeated  throughout  the  work,  as 
appertaining  to  the  whole — the  Newtonian  mechanics  and  the  doctrine  that 
heat  and  work  are  one.  But,  in  oral  discussion  of  these  matters  with  those 
who  are  entitled  to  speak  as  physicists,  he  has  frequently  met  the  objection 
stated,  that  it  was  impossible  to  explain  the  gigantic  forces  and  energies 
of  molecular  mass  mechanically.  But  in  every  case,  upon  investigation,  it 
has  developed  that  this  position  has  been  based  upon  some  quite  unwar- 
ranted and  limiting  assumption — such  as  this  that  molecular  mass  must  b° 
of  densities  similar  to  mundane  density — which  is  in  reality  gratuitous 
and  unnatural.  It  is  to  show  that  such  assumptions  are  not  only  need- 
less, but  groundless,  that  this  argument  has  been  inserted. 


128  ENERGY 

constitute,  when  viewed  in  proportion  to  the  linear  distances  and 
volumes  also  employed,  a  most  extreme  condition  of  concentra- 
tion of  matter.  If  these  engineering  bodies  were  to  be  located 
properly  on  the  field  of  Fig.  8,  or  Fig.  12,  for  instance,  their 
place  would  be  found  at  the  extreme  prolongation  of  the  curve 
CB  beyond  B.  That  is  to  say,  iron,  steel  and  granite,  as  used  in 
machine  construction,  are  about  as  far  removed  from  their 
natural  mean  energetic  conditions  as  ice,  when  at  a  temperature 
of  — 400°  Fahr.,  is  removed  from  that  condition  of  boiling  water 
or  superheated  steam  wherein  it  displays  its  greatest  thermo- 
dynamic  adaptability  for  handling  large  quantities  of  energy, 
both  in  thermogy  and  labority — the  condition  in  which  it  explodes 
our  boilers,  drives  our  engines  and  ameliorates  the  sudden 
changes  of  climate  and  season. 

It  now  appears  that  a  finally  exact  definition  of  heat  is  a 
very  difficult  thing.  In  the  Seventh  Paper  it  was  defined  as  the 
spacial  and  kinetic  relativities  between  the  particles  of  a  body, 
excluding  the  subpermanent  or  colliding  particles.  Now  it  seems 
that  the  definition  of  even  so  simple  a  term  as  collision  is  difficult. 
Collision  is  an  approach  so  close  as  to  upset  the  permanence  of 
equilibrium  of  molecular  existence.  A  few  years  ago  we  should 
have  said  that  this  settled  it :  We  knew  what  permanence  meant. 
Now  questions  are  raised  as  to  the  permanence  of  existence  of 
chemical  matter,  by  the  degradation  of  radioactive  matter,  which 
are  hard  to  answer;  and  heat  can  scarcely  be  a  more  permanent 
form  of  energy  than  chemical  energy. 

Moreover,  little  light  has  yet  been  shed  upon  the  question 
whether  the  superpermanent,  or  hyperbolic,  energies  of  the 
molecule  should  be  included  as  heat.  To  the  writer,  they  plainly 
should. 

It  will  be  found  of  great  assistance  to  the  understanding  of 
heat-action,  in  the  next  paper,  even  if  we  do  not  know  just 
what  heat  is,  to  have  done  these  things,  viz:  (i)  To  have  dis- 
posed of  the  idea  of  the  "perfectly  elastic  yet  solid"  molecule; 
(2)  to  have  reduced  the  definition  of  contact  and  collision  to 
their  proper  places;  (3)  to  have  similarly  disenthroned  the 
usurper  called  "perfect  gas";  and  (4)  to  have  established  heat- 
action  similarly  with  mechanical  action  upon  the  basis  of  the 
mean  energetic  condition,  depending  upon  action  at  a  distance, 
rather  than  upon  that  natural  unreality,  contact. 


CHAPTER  XI. 

THE  Two  BASIC  THERMAL  PROCESSES. 

HEAT-TRANSFER  AND  WORK-PERFORMANCE. 

Chapter  IX  announced  the  need,  in  thermodynamic  discussion, 
of  mechanical  concepts  of  pressure,  volume,  temperature  and 
entropy.  The  first  two  of  these  obscure  properties  of  matter 
have  now  been  discussed,  in  their  mechanical  aspect.  Before 
proceeding  to  a  similar  discussion  of  temperature  and  entropy, 
however,  it  will  be  necessary  to  develop  the  mechanical  concepts 
of  the  two  basic  thermodynamic  processes,  heat-transfer  and 
work-performance,  as  contrasted  with  static  thermal  attributes, 
such  as  pressure,  volume,  temperature  and  entropy. 

The  thermal  diagram,  Fig.  8,  which  was  presented  in  Chapter 
VIII  (page  100),  developed  before  the  eye  only  two  out  of  the 
several  simple  thermal  processes  which  are  familiar  to  boiler 
and  engine-room.  These  two  were,  first,  the  isomorphic  and 
metathermal*  heating  and  cooling  of  bodies.  This  process  was 
explained  as  if  it  were  performed  only  by  heat  developed  by 
impact  and  friction ;  but  heat  supplied  by  conduction  or  radiation 
would  have  been  found  to  produce  identical  results.  The  second 
process  was  the  mctamorphic  and  isothermal  one  of  addition  or 
abstraction  of  heat  when  the  body  was  neither  heated  nor 
cooled,  but  changed  its  physical  state  instead,  as  in  fusion  or 
vaporization.  This  too  was  explained  with  heat  furnished  by 
impact  or  friction,  although  heat  produced  by  conduction  or 
radiation  would  have  produced  the  same  results.  In  the  latter 
case,  however,  the  original  form  of  the  energy  would  have  been 
as  obscure  to  the  understanding  as  the  final  one  (whereas  we 
feel  that  we  understand  the  mechanical  nature  of  impact  or 
friction)  and  so  would  have  been  of  little  aid  to  the  under- 
standing. 

The  familiar  thermal  phenomena  of  the  power-house,  how- 
ever, include  at  least  two  other  processes  which  need  explanation. 

*Meaning  "temperature-changing,"  contrasted  with  isothermal.  "Meta" 
signifies  change,  as  "iso"  signifies  constancy. 

129 


130  ENERGY 

These  are,  first,  adiabatic  work-performance  by  heat,  and,  sec- 
ondly, the  isenergic  degradation  of  heat  by  wire-drawing.  It  will 
be  assumed  that  the  reader  is  sufficiently  familiar  with  both  of 
these  processes  to  avoid  the  necessity  of  their  general  descrip- 
tion. However,  in  order  to  make  prominent  the  features  which 
are  sufficiently  characteristic  to  aid  in  understanding  their  internal 
mechanical  operation,  they  will  be  briefly  defined. 

The  Adiabatic  Process.  Adiabatic  action  occurs  only  when 
a  body  undergoes  simultaneous  change  in  volume  and  tem- 
perature, while  expanding  or  being  compressed,  with  the  exclusion 
of  heat-interchange  by  conduction  or  radiation  or  friction  or 
impact  with  other  bodies.  The  only  permissible  exchange  of 
energy  with  the  outside  world  is  in  the  form  of  work;  and  this 
exchange  is  unavoidable,  if  the  volume  is  to  alter,  for  pressure 
exists  everywhere. 

The  energy  thus  exchanged  is  supplied  from  the  body's  fund 
of  heat;  and  not  only  from  its  fund  of  heat,  but  from  its  fund 
of  temperature-heat.  For  no  other  sort  of  heat  is  available  for 
work-performance.  Not  only  does  adiabatic  work-performance 
always  result  in  temperature-change,  but  it  is  the  only  known 
exact  measure  of  temperature-change.  This  fact  is  the  basis  of 
Carnot's  foundation  of  thermodynamics,  and  of  Lord  Kelvin's 
perfect  scale  of  temperature,  in  which  equal  "degrees"  are  defined 
in  terms  of  equal  amounts  of  work  performed,  not  in  terms  of 
equal  amounts  of  heat  transferred ;  nor,  as  in  our  ordinary  ther- 
mometry,  in  terms  of  equal  increments  in  volume  of  expansion. 

In  adiabatic  action  the  alterations  in  volume  and  temperature 
occur  in  inverse  directions.  The  temperature  decreases  as  the 
volume  increases,  and  vice  versa.  The  process  is  represented  in 
the  thermal  diagram,  Fig.  8,  by  a  straight  vertical  line. 

The  Wire-drawing  Process.  When  a  fluid,  whether  gas- 
eous or  liquid,  flows  through  a  pipe  or  orifice,  the  flow  is  resisted 
by  friction.  The  thermodynamic  effect  of  this  takes  either  one 
of  two  forms. 

I.  If  the  pressure  upon  the  substance  be  greater  than  its 
vapor-tension,  for  all  the  temperatures  in  question,  the  tem- 
perature of  the  fluid  will  rise  and  its  pressure  fall  as  it  proceeds. 
This  process  is  an  isomorphic  one  of  heating  by  friction  and 
impact,  as  already  fully  described,  except  that  it  occurs  under 


TWO  BASIC  THERMAL  PROCESSES  131 

falling  pressure.  Such  is  the  action,  for  instance,  in  all  water- 
pipes  of  ordinary  temperature. 

2.  If  the  pressure  at  any  point  should  become  less  than  the 
vapor-tension  corresponding  to  the  local  temperature  (and  it  is 
obvious  that  Process  No.  I  tends  to  result  in  this  state  of  affairs, 
if  carried  far  enough),  then  the  wire-drawing  process  ceases  to 
be  isomorphic  and  becomes  metamorphic.  The  substance  begins 
to  vaporize.  Such,  for  instance,  is  the  action  in  the  ordinary 
boiler-blow-off,  particularly  if  the  blow-off  pipe  be  imagined  as 
so  long  that  virtually  all  of  the  energy  of  the  escaping  fluids  is 
spent  in  overcoming  friction.  The  same  action  occurs  in  the 
expansion-valve  of  the  ammonia  refrigerating-machine. 

Or,  if  the  substance  happens  to  be  already  vaporized  when  it 
starts  upon  its  impeded  flow,  as  is  the  case  with  steam  flowing 
through  a  steam-pipe,  then  the  result  of  the  isenergic  action  will 
be  to  further  rarefy  it.  This  action  is  familiar  in  the  ordinary 
throttling  and  superheating  calorimeter. 

Between  these  two  numerated  forms  of  isenergetic  wire- 
drawing there  exists  a  most  instructing  similarity  and  contrast. 
The  similarity  lies  in  the  fact  that  both  processes  increase  the 
quantity-factor  of  the  heat,  or  its  entropy.  The  contrast  lies  in 
the  fact  that  whereas  the  first  process  increases  the  temperature, 
the  second  decreases  it. 

The  conclusions  to  be  drawn  are  obvious,  and  are  two-fold. 

1.  Friction  and  impact  are  intimately  and  functionally  linked 
with  entropy-change. 

2.  Frictional  or  impactive  increase  of  entropy  occurs  inde- 
pendently of,  and  has  no  determinative  effect  upon,  temperature. 
The  question  as  to  whether  the  temperature  is  to  rise  or  fall  or 
remain  constant,  as  the  result  of  friction  or  impact,  depends 
solely  upon  the  external  pressure.    For,  as  has  just  been  pointed 
out,  the  increase  of  entropy  by  impact  or  friction  under  rising, 
falling    or    stably    constant   temperature    are    processes    equally 
familiar  in  nature. 

In  order  to  be  sure  of  understanding  this  position,  it  were 
well  to  examine  further  the  nature  of  wire-drawing. 

Taking,  for  convenience,  the  second,  or  expansive,  form  of 
wire-drawing,  it  soon  reveals  itself  as  a  duplex  process.  It  con- 
sists, in  reality,  of  two  distinct  processes  merged  into  an  appar- 
ent, but  not  real,  unity.  First  comes  the  acceleration  of  the  mass 


132  ENERGY 

of  liquid  or  gas  into  linear  motion.  This  absorbs  work;  and  as 
the  only  source  of  energy  for  this  purpose  is  the  body's  heat,  the 
temperature  falls  to  supply  it.  This  part  of  the  wire-drawing 
process  is  purely  adiabatic. 

But  the  linear  motion  of  the  particles  is  no  more  than  gene- 
rated than  it  engenders  friction;  and  the  word  friction  is  merely 
our  short  name,  as  has  been  seen,  for  the  destruction  of  linear 
motion  in  the  increase  of  the  quantity-factor  of  heat.  In  the 
present  case,  since  the  rate  of  flow  is  in  equilibrium  with  the 
friction  opposing  it,  the  rate  of  energy-transformation,  all  around, 
is  controlled  by  the  degree  of  friction  present.  The  rough,  hard 
walls  turn  the  work  of  flow  into  heat  as  fast  as  they  can,  and 
the  consequent  deficit  in  mechanical  energy  is  made  good  con- 
tinuously by  the  further  adiabatic  expansion  of  the  body. 

Now  it  happens  that  wire-drawing  is  our  only  instance  of 
simultaneous  adiabatic  expansion  and  friction.  The  steam- 
turbine  seems  at  first  to  offer  another  and  more  illustrative  case. 
But  upon  second  thought  it  appears  that  its  internal  action  is  the 
same  as  pure  wire-drawing,  except  that  it  is  complicated  by  the 
simultaneous  presence  of  still  a  third  process,  viz:  adiabatic 
work-performance  upon  outside  systems.  In  steam-engine  cyl- 
inders we  have  an  obverse  illustration,  viz:  the  addition  or 
abstraction  of  heat  while  adiabatic  work-performance  is  going 
on.  But  to  bring  this  into  line  with  pure  wire-drawing  we  must 
introduce  the  inference,  which  is  everywhere  forced  upon  us, 
that  friction  and  thermal  conduction  are  identical  processes,  in 
their  ultimate  nature;  except  that  whereas  conduction  will  work 
both  ways — in  and  out — friction  will  work  only  one. 

Nevertheless,  these  facts  serve  only  to  broaden,  rather  than 
to  narrow,  the  conclusions  stated  above,  viz : 

•I.  That  friction  affects  only  entropy,  being  indeterminative 
of  temperature-change ; 

2.  That  the  direction  of  temperature-change  depends  solely 
upon  whether  there  be  a  surplus  of  external  over  internal  pres- 
sure, radially  active,  or  not ;  and 

3.  That   in  the  apparently   single   process  of  wire-drawing 
there  really  exists  a  merger  of  both  the  above  independent  proc- 
esses, viz :   a  variation  of  the  entropy  by  friction,  and  a  variation 
of  the  temperature  by  pressure-action  or  work-performance. 

If  we  turn  now  from  the  development  of  heat  by  friction  or 


TWO  BASIC  THERMAL  PROCESSES  133 

impact  to  its  twin  brother,  the  development  of  heat  by  thermal 
conduction,  there  appears  a  most  striking  similarity  to  the  wire- 
drawing situation.  That  is  to  say,  not  only  does  thermal  con- 
duction identify  itself,  by  ways  not  necessarily  repeated  here, 
with  friction  and  impact  as  an  inevitable  controller  of  entropy- 
change,  but  the  temperature-change  which  accompanies  it  is  not 
determined  by  the  quantity  of  heat  conducted,  but  solely  by  the 
relation  between  external  and  internal  pressures.  If  the  sub- 
stance be  water,  for  instance,  and  if  the  pressure  upon  it  be 
greater  than  its  vapor-tension,  then  the  addition  of  heat  by 
thermal  conduction,  or  by  friction  either,  increases  the  tem- 
perature— or  rather,  it  results  in  an  increase  in  temperature. 
But  if  the  external  pressure  and  the  vapor-tension  be  balanced, 
as  is  the  case  in  the  ordinary  steam-boiler,  then  the  addition  of 
heat  by  thermal  conduction,  or  by  friction  either  (though  in  this 
case  we  lack  a  familiar  instance),  results  in  no  rise  in  tem- 
perature. The  water  merely  vaporizes  isothermally. 

If,  again,  the  external  pressure  happens  to  be  less  than  the 
vapor-tension — as  is  the  case  in  a  steam-engine  cylinder  when 
the  resistance  of  the  piston  is  less  than  the  expansive  force  of 
the  steam — then  the  addition  of  heat  by  thermal  conduction, 
such  as  occurs  if  the  engine-cylinder  be  steam- jacketed,  occurs 
in  connection  with  a  fall  of  temperature.  Or  the  wire-drawing 
of  steam,  already  described,  is  an  even  better  illustration. 

We  have  finally  come  to  the  point,  therefore,  where  the 
absurdity  of  calling  temperature-change  the  effect  of  either 
friction  or  thermal  conduction  has  become  obvious.  It  is  entropy- 
change  alone  which  is  this.  Temperature-change  must  be  the 
functional  result  of  something  quite  different.  In  cylinder- 
expansion  we  need  no  assurance  that  it  is  the  deficit  of  pressure, 
with  the  consequent  retreat  of  the  confining  walls,  which  creates 
the  drop  in  temperature.  The  simultaneous  addition  of  heat  has 
nothing  to  do  with  the  case,  except  in  the  alteration  of  the 
entropy  present.  But  in  the  addition  of  heat  to  the  water  in  a 
boiler,  thermal  conduction  has  no  more  to  do  with  the  tem- 
perature-m^  than  it  has,  in  the  steam- jacketed  cylinder,  with  the 
the  temperature-/a//.  It  is  pressure  alone  which  controls  both. 

It  is  only  because  the  great  majority  of  all  the  things  to 
which  we  have  been  accustomed  to  add  heat  have  had  vapor- 
tensions  below  that  of  the  surrounding  atmosphere,  or  other 


134  ENERGY 

external  pressure,  that  we  have  come  to  associate  the  addition 
of  energy  by  thermal  conduction,  or  by  friction  and  impact,  with 
rise  of  temperature.  This  popular  association  of  these  two 
phenomena  is  not  only  loose  and  inaccurate,  but  it  is  funda- 
mentally erroneous  and  misleading.  It  begets  in  the  student's 
mind  an  idea  of  interdependence  between  temperature-change 
and  thermal  conduction  which  is  in  direct  antithesis  to  the 
principle  laid  down  by  Lord  Kelvin,  viz :  that  the  only  correct 
measure  of  temperature-change,  the  only  correct  thermometric 
degree,  is  work-performance.  It  draws  the  very  foundation 
from  beneath  what  should  be  the  first  and  fundamental  idea 
taught  the  student  in  the  thermal  laboratory,  viz:  that  thermal 
conduction  has  everything  to  do  with  alterations  in  entropy,  and 
nothing  whatever  to  do  with  alterations  in  temperature. 

The  deliberate  association  of  thermometry  and  thermal  con- 
duction in  the  physical  laboratory  should  cease.  The  first  efforts 
of  the  instructor  in  thermodynamics,  after  a  clear  concept  of  the 
general  nature  of  energy  is  once  implanted,  should  be  directed 
toward  a  breaking  up  of  this  false  association  of  ideas,  already 
too  deeply  imbedded  by  the  ordinary  experiences  of  life  before 
reaching  the  study  of  the  natural  sciences. 

ONLY  TWO  FUNDAMENTAL  THERMODYNAMIC  PROCESSES. 

The  conclusion  is  inevitable,  therefore,  that  there  exist  but 
two  basic,  elementary,  thermodynamic  processes.  One  of  these 
is  the  generation  of  thermal  quantity-factor,  or  entropy,  by 
impact,  friction,  thermal  conduction  or  any  other  process  which 
proves  upon  examination  to  be  an  equivalent:  a  process  which 
always  occurs  at  constant  temperature,  or  independently  of  and 
ineffectively  upon  the  temperature. 

The  other  process  is  the  adiabatic  variation  of  temperature 
by  work-performance :  a  process  which  always  occurs  at  constant 
entropy,  or  independently  of  and  ineffectively  upon  entropy- 
change,  but  which  cannot  occur  independently  of  temperature- 
change. 

In  order  that  the  fundamental  importance  of  these  two  proc- 
esses may  be  clearly  brought  out,  they  may  be  given  distinctive 
names  and  completely  defined,  as  follows : 

i.  THERMOGY:  the  isothermal  variation  of  entropy,  occur- 
ring always  when  matter  is  subjected  to  impact,  friction,  radia- 


TWO  BASIC  THERMAL  PROCESSES  135 

tion,  thermal  or  electrical  conduction,  or  perhaps  other  processes 
as  yet  unidentified.  In  the  mathematical  discussion  of  thermogy 
the  temperature  T  must  always  appear  as  an  integer,  and  the 
entropy-change  dN  as  a  differential. 

2.  LABORITY:  the  isentropic,  or  adiabatic,  variation  of  tem- 
perature, occurring  always  when  matter  develops  work  from 
heat,  and  sometimes  when  matter  develops  heat  from  work— that 
is,  when  it  absorbs  work  radially  with  perfect  elasticity,  or 
"reversibly."  In  the  mathematical  discussion  of  labority  the  tem- 
perature-factor dT  must  always  appear  as  a  differential,  and  the 
entropy- factor  N  as  an  integer. 

The  first  of  these  processes  must  always  be  represented  in 
the  thermal  diagram,  Fig.  8,  by  a  straight  horizontal  line,  prop- 
erly of  infinitesimal  length,  but  integrable,  under  proper  conditions, 
into  a  finite  extent.  The  second  process  must  be  represented  by 
a  straight  vertical  line,  under  similar  conditions. 

All  other  thermodynamic  processes  whatever,  of  which  several 
are  commonly  depicted  upon  the  thermal  diagram  by  various 
oblique  and  curved  lines,  must  be  considered  as  made  up,  in 
natural  fact  as  well  as  geometrically,  of  these  two  basic  com- 
ponents, occurring  simultaneously  and  combining  to  produce  an 
apparently  single,  though  really  double,  result. 

Thus,  in  the  isomorphic  heating  of  water  under  pressure, 
which  is  represented  by  the  oblique  curve  ADW,  Fig.  8,  the 
energy  is  supplied  in  such  form  (either  by  friction,  impact,  con- 
duction or  radiation)  that  the  quantity-factor  is  increased  by  the 
value  dN,  say.  The  quantity  of  energy  thus  supplied,  dQ,  is 
equal  to  T  dN  and  is  represented  by  a  vertical  element  of  area 
between  ADW  and  OX.  The  result  of  this  supply  of  energy  is 
to  vaporize  momentarily  a  minute  quantity  of  the  water. 

If  the  external  pressure  upon  the  water  be  exactly  balanced 
with  the  vapor-tension,  as  along  DE,  this  minute  portion  of 
vapor  remains  vapor;  and  further  increments  of  vapor,  from 
further  increments  of  heat,  also  remain  vapor,  cumulatively. 
But  if  the  external  pressure  be  in  excess,  as  is  necessary  for 
the  prosecution  of  the  isomorphic  AD  toward  D',  the  tiny  bits 
of  vapor  are  recompressed,  adiabatically,  into  water,  as  fast  as 
they  are  produced,  leading  to  the  adiabatic  rise  in  temperature 
dT.  The  result  of  this  simultaneous  increase  of  quantity-factor 
and  temperature  is  portrayed  by  the  curve  DW. 


136  ENERGY 

The  other  isomorphic  curves  might  be  similarly  explained, 
except  that  in  the  case  of  ice  the  reaction  cannot  occur  between 
solid  and  liquid  forms,  because  pressure  turns  ice  to  water,  not 
water  to  ice.  The  only  way  out  of  the  difficulty  is  to  assume 
that  there  must  be  present  in  the  ice,  at  all  temperatures,  minute 
quantities  of  both  water  and  water-vapor,  between  which  the 
reaction  occurs;  and  this  fact  has  been  developed  independently 
by  direct  observation.  In  the  case  of  refractory  solids,  which 
refuse  to  fuse  under  all  ordinary  applications  of  heat,  it  is  to  be 
remarked  that  the  resistance  to  fusion  is  plainly  involved  in  the 
crystalline  form  of  the  solids — a  question  too  complex  and 
bearing  too  lightly  upon  power-engineering  for  discussion  here. 

Throughout  the  previous  discussion  the  quantity-factor  of 
heat  has  appeared  as  a  matter  of  fundamental  import.  Obvi- 
ously, it  also  needs  a  short  and  suitable  name.  Hitherto  it  has 
been  referred  to  often  merely  as  "the  quantity-factor,"  in  order 
that  the  discussion  might  not  be  vitiated  by  doubts  as  to  the 
propriety  of  a  name,  or  by  preconceived  notions  as  to  the  nature 
of  entropy,  the  only  name  suitable  for  it.  But  the  analysis  has 
now  reached  a  stage  where  these  considerations  need  no  longer 
defer  the  identification  of  the  quantity-factor  of  Fig.  8  with  the 
term  entropy  which  was  invented  by  Professor  Clausius  forty 
years  ago.  Fig.  8  itself  may  be  similarly  identified  with  the 
entropy-temperature  diagram  of  Gibbs  and  Macfarlane  Gray. 

It  should  be  explained  immediately  that  this  identification  is 
not  complete  nor  exact  nor  final;  but  this  is  so  only  because 
doubt  exists  as  to  the  natural  facts  upon  which  the  definition 
of  entropy  rests.  Professor  Clausius  defined  his  understanding 

of  entropy  as  being  a  function  "equal  to  or  greater  than,  but 

jr\  » 
never  less  than,  the  integration  of  — -^—  in  any  thermodynamic 

cycle.  The  equality  was  to  apply  when  the  cycle  is  free  from 
impact  and  friction,  or  was  "reversible."  The  increase  in  entropy 
would  occur  when  these  were  present — as  they  always  are. 

But  Clausius's  definition  of  entropy,  while  deserving  the  very 
great  credit  of  being  the  first  to  appear,  is  nevertheless  incom- 
plete. It  is  incomplete  in  regarding  itself  as  complete  and  final. 
It  is  incomplete  in  three  distinct  ways. 

First,  while  Clausius's  definition  of  entropy  is  exact  enough 
in  terms  od  dQ,  he  gives  no  definition  of  dQ.  None  of  the  text- 


TWO  BASIC  THERMAL  PROCESSES  137 

books  which  have  followed  Clausius  attempt  to  define  it.  We 
are  still  to-day,  although  possessing  a  vastly  greater  accumulation 
of  data  than  in  Clausius's  time,  unable  to  define  dQ.  Clausius 
defined  dQ  in  terms  of  impact,  friction,  radiation  and  conduction. 
We  know  now  that  that  list  is  incomplete •;  but  what  to  add  to  it 
(beyond  some  types  of  chemical,  electrical  and  magnetic  action), 
and  how  we  are  to  know  when  we  have  finished  the  list,  we  are 
yet  unable  to  say. 

Secondly,  Clausius  recognized  no  other  form  of  entropy  than 
thermal  entropy.  In  Clausius's  day,  although  the  Conservation 
of  Energy  had  been  foreshadowed  by  Newton  (1680)  and 
defined  and  pictured  pretty  completely  in  the  writings  of  Mohr 
(1837,  Mayer  (1843),  Grove  (1846)  and  Helmholz  (1847), 
two  decades  before  Clausius,  yet  the  idea  that  heat  is  a  mode 
of  motion  was  still  obscure  in  the  scientific  world.  It  was  far 
beyond  the  natural,  for  that  day,  for  Clausius  to  see  that  when 
he  had  identified  entropy-production  with  impact  and  friction, 
which  are  purely  mechanical,  he  had  proven  that  there  must  be  a 
mechanical  form  of  entropy,  even  if  there  were  none  of  heat  or 
temperature.  Thus,  long  before  the  brilliant  work  which  began 
with  Count  Rumford  and  ended  with  Kelvin,  Joule  and  Maxwell 
had  settled  beyond  peradventure  the  physical  reality  and  the 
mechanical  nature  of  both  heat  and  temperature,  the  real  (albeit 
undeveloped)  foundation  for  the  identification  of  entropy  also,  as 
a  physical  reality  and  a  feature  of  mechanical  energy,  had  been 
laid,  in  the  original  discovery  of  the  function  itself. 

To-day  the  doubt  lies  not  as  to  whether  entropy  be  a  physical 
reality,  such  as  temperature,  heat  or  work,  rather  than  a  mere 
mathematical  function.  It  does  not  even  lie  in  the  direction  of 
whether  there  may  be  other  sorts  of  entropy  than  thermal  entropy. 
Our  difficulty  to-day  is  to  know  where  the  concept  of  entropy,  as 
a  physical  reality,  is  to  end.  We  know  that  as  far  as  energy- 
transformations  extend,  so  far  do  all  of  the  forms  of  energy 
exhibit  the  dual  attributes  of  intensity  and  extensity,  or  quantity- 
factor,  respectively.  If  the  energy  itself  is  identical  throughout 
all  these  changes,  so  must  be  the  two  component  factors.  There 
must  be  as  many  forms  of  temperature  and  entropy  as  there  are 
forms  of  energy  interchangeable  with  heat.  But  for  their  exact 
definition  we  must  await  further  knowledge. 

Thirdly,   Clausius   saw   no   chance   for  the   entropy  of   any 


138  ENERGY 

system  ever  to  decrease.  Knowing  no  other  form  of  entropy 
than  thermal  entropy,  this  was  most  natural ;  for  thermal  entropy, 
so  long  as  it  stays  thermal,  does  tend  constantly  to  increase — 
upon  the  earth's  surface,  at  any  rate.  But  more,  so  far  as 
scientific  thought  may  surmise,  everywhere  in  nature  exists  some 
solidity,  or  at  least  viscosity;  and  wherever  solidity  or  viscosity 
exists,  mechanical  motion  is  being  constantly  converted  into  heat 
and  heat  is  degrading  itself  in  temperture,  with  increase  of 
entropy. 

Starting  from  these  facts  Professor  Zeuner  announced  his 
theorem  that  the  entropy  of  the  universe  tends  always  to  a 
maximum.  Proceeding  further  from  this  same  base,  Lord 
Kelvin  gave  countenance  to  the  cosmic  doctrine  of  the  general 
degradation  of  all  forms  of  energy  into  unavailability;  and  the 
school  of  what  is  now  orthodox  thermodynamics  was  settled 
beyond  discussion.*  Whether  there  be  such  a  thing  or  not  as  a 
universal  degradation  of  energy  will  be  discussed  in  a  later 
chapter.  The  writer  may  announce  here  that  he  does  not  believe 
that  there  is.  He  believes  that  the  availability  of  the  energy  of 
the  universe  remains  constant,  although  in  regions  as  great  as  a 
solar  system  it  may  locally  and  temporarily  fall  into  deficit  or 
accumulate  into  a  surplus. 

But  it  is  not  upon  any  principle  so  broad  as  this  that  the 
concept  of  entropy  must  rest.  It  rests  upon  the  familiar,  every- 
day transformations  of  energy,  and  upon  the  fact  that  therein 

*The  following  extract  is  taken  from  a  lecture  by  Sir  Oliver  Lodge 
upon  Lord  Kelvin,  which  was  delivered  after  these  pages  had  been  written : 

"I  fancy  that  he  himself,  and  certainly  some  of  his  disciples,  have 
been  at  times  inclined  to  attribute  to  the  law  of  degradation  mort  ulti- 
mate and  cosmic  importance  than  properly  belongs  to  it.  Its  significance 
is  limited  to  the  validity  of  the  terms  'heat'  and  'temperature';  and 
if  for  any  reason  these  cease  to  have  a  practical  meaning,  then  the  dissi- 
pation of  energy  ceases  to  be  inevitable.  *  *  The  different  availabilities 
of  various  kinds  must  be  essentially  a  human  and  temporary  conception, 
useful  and  convenient  for  practical  purposes,  but  not  ultimate  nor 
cosmic.  *  *  The  dissipation  of  energy  has  no  meining  when  'heat' 
and  'temperature'  are  obsolete  terms ;  that  is  to  say,  when  what  we  now 
consider  to  be  unorganized  and  intractable  molecular  motions  can  be  dealt 
with  in  an  individual  and  organized  way." 

Professor  Zeuner,  too,  is  to  be  credited  with  the  first  perception  of  the 
physical  reality  of  entropy.  His  term  for  it,  "heat-weight,"  is  a  most 
graphic  expression  of  one  of  the  prime  characteristics  of  entropy,  namely, 
its  "heft"  or  forcefulness  in  work-performance.  But  entropy,  we  shall 
see,  is  not  the  force  of  thermal  gravitation,  but  the  degree  of  subdivision 
of  matter  which  develops  that  force,  and  which,  taken  with  propinquity, 
determines  its  magnitude. 


TWO  BASIC  THERMAL  PROCESSES  139 

appears  everywhere  this  duality  of  intensity-factor  and  quantity- 
factor  of  energy,  running  through  all  the  natural  sciences.  Its 
importance  is  basic,  not  only  in  the  cases  of  heat  and  work, 
which  alone  can  be  discussed  here,  but  in  chemical,  electrical, 
magnetic  and  radiant  energies  as  well.  Although  it  is  only  in 
mechanical  and  thermal  energies  that  the  quantity-factor  has 
been  accurately  defined — as  S  (M^MJ  or  the  extent  of  mass- 
pairing  present  in  the  first,  and  as  entropy  in  the  second — yet  in 
all  the  other  sciences  its  existence  and  importance  are  none  the 
less  indubitable  because  vague.  In  all  these  fields  it  demands 
immediate  definition.  Not  only  must  the  engineer,  the  electrician 
and  the  chemist  contribute  to  the  physicist  their  data,  before  this 
general  energy-factor  which  in  heat  finds  the  name  entropy  can 
be  properly  defined,  but  the  biologist,  the  economist  and  the 
sociologist  must  follow  suit.  In  all  of  these  sciences  there  has 
been  too  much  time  devoted  to  study  and  measurement  of  the 
visible  space-and-motion  factors,  and  far  too  little  attention  given 
to  understanding  what  it  is  which  can  experience  space  or 
motion.  There  need  be  no  hesitancy  in  stating  that  when  our 
students  are  properly  taught  the  nature  of  entropy,  in  its  broadest 
sense,  they  will  find  the  concept  of  equal  use,  in  after  life,  in 
their  science,  their  engineering,  their  business  and  their  politics.* 
For  all  these  reasons  the  writer  will  use  the  term  entropy  in 
a  considerably  wider,  and  a  somewhat  different,  sense  from  that 
given  it  originally  by  Professor  Clausius  (if  a  bare  and  rigid 
mathematical  definition  can  be  called  a  "sense").  He  will  use 
it  as  synonymous  with  the  quantity-factor  of  heat,  and  he  will 
show  in  the  next  paper  that  it  is  identical  with  the  quantity- 
factor  in  mechanical  energy,  S  (M^M.,).  He  will  also  suggest 
briefly  the  idea  that  this  same  quantity-factor  runs  through  all 
the  other  forms  of  energetic  action  to  which  the  laws  of  the 
transformation  and  conservation  of  energy  apply,  in  the  inani- 
mate, vegetable,  animal,  human  and  social  activities  of  the  uni- 
verse. It  will  appear  that  no  student  may  lay  claim  to  a  grasp 
of  the  fundamental  principles  which  guide  and  control  any  of 
these  activities,  without  first  acquiring  a  thorough  comprehension 
of  the  mass-factor  in  mechanical  energy — that  it  is  the  form  of 

*The  reader  who  desires  a  further  development  of  this  idea  will 
find  its  discussion  in  the  Harvard  Engineering  Journal  for  1008,  January 
and  November  issues.  The  broader  significance  of  the  quantity- factor  of 
energy  receives  further  discussion  in  the  last  chapter  of  this  book. 


140  ENERGY 

association,  or  grouping,  or  relationship,  between  things,  and  not 
the  things  themselves,  which  gives  rise  to  energy,  and  to  the 
characteristics  of  its  results. 

The  next  thing  is  to  understand  this  corresponding  thing  in 
heat  which  we  call  entropy.  The  final  fruit  is  to  comprehend 
that  all  forms  of  energetic  activity,  including  social  energetics, 
are  based  upon  the  same  factors  and  follow  the  same  laws  as 
those  which  support  and  guide  the  silent  stars,  the  incandescent 
flame  and  the  awful  thunderbolt. 


CHAPTER  XII. 

MECHANICAL  CONCEPTS  OF  THERMAL  PHENOMENA. 
B.   TEMPERATURE  AND  ENTROPY. 

The  definitions  given  in  the  preceding  paper  for  the  two 
basic  thermodynamic  processes,  thermogy  and  labority,  now  pave 
the  way  for  clear  mechanical  concepts  of  temperature  and 
entropy.  Before  attempting  to  formulate  any  such  concepts, 
however,  it  were  well  to  get  well  in  mind  the  peculiarities  which 
are  characteristic  of  the  two  quantities  respectively. 

Temperature.  The  prime  characteristic  of  temperature,  and 
the  only  true  measure  of  temperature-change,  is  work-perform- 
ance. Work  can  be  performed  at  the  expense  of  heat  only 
when  the  temperature  falls.  Work  can  be  absorbed  elastically, 
with  obvious  conservation  of  energy,  or  "reversibly"  as  it  is 
called,  only  when  the  temperature  rises. 

This  idea  of  temperature  we  owe  originally  to  Carnot  (1825), 
who  stated  the  law  that  the  work  performed — if  the  process 
were  perfectly  elastic,  or  ''radial,"  or  "reversible" — was  pro- 
portional to  the  temperature-fall;  that  is,  the  law  is  conditioned 
upon  pure  labority,  devoid  of  thermogy — the  straight,  vertical, 
adiabatic  process.  Half  a  century  later  this  idea  was  amplified 
by  Lord  Kelvin  into  the  principle  that  the  only  accurate  measure 
of  temperature-variation,  or  the  only  true  thermometric  scale, 
was  to  be  based  upon  equal  degrees,  not  of  heat-supply,  nor  of 
volume  of  expansion,  but  of  zvork-performance. 

The  second  characteristic  of  temperature  is  the  universal 
tendency  of  differences  of  temperature  to  set  up  energy- transfer 
by  thermal  conduction  and  radiation.  This  is  the  explanation  of 
the  familiar  sensations  of  hot  or  cold — the  gain  or  loss  of 
energy  by  the  skin  and  nerves  when  brought  into  contact  with 
other  bodies,  or  when  exposed  to  radiation.  These  ideas  of 
temperature  long  antedated  the  discovery  of  the  work-scale  of 
temperature;  yet  they  are  by  no  means  such  accurate  guides  in 
the  understanding  of  the  nature  of  heat.  They  must  be  con- 

141 


142  ENERGY 

sidered  quite  secondary  to  the  prime  characteristic  of  tem- 
perature-heat :  work-performance. 

These  sensations  are  deceptive  because  they  concern  merely 
the  rate  of  transfer  of  energy,  a  thing  which  is  subject  to  other 
influences  than  mere  temperature-difference.  Thus,  exposure  to 
the  wind,  or  wetting  the  skin,  will  heighten  the  sensation  of 
cold.  The  latter  of  these  two  processes  may  actually  lower  the 
temperature  and  be  perceived  by  the  thermometer ;  but  the  former 
makes  no  difference  to  the  thermometer. 

A  still  more  accidental  characteristic  of  temperature  is  its 
effect  upon  the  volume  of  a  substance.  All  gases  and  vapors 
increase  in  volume  with  rising  temperature.  Some  liquids  and 
solids  do  the  same,  though  at  a  lower  rate ;  but  many  reverse  the 
rule  and  increase  in  volume  with  falling  temperature.  Yet  this 
characteristic  of  temperature  is  our  main  reliance  in  ordinary 
thermometry.  The  relative  expansion  of  mercury  and  glass  is 
the  most  common  means  for  exhibiting  temperature-changes. 
For  more  accurate  purposes  hydrogen  is  substituted  for  the 
mercury. 

A  similar  characteristic  of  temperature  is  the  proportionality 
to  it  of  the  expansive  pressure  of  gases  and  vapors.  This 
proportionality,  like  that  of  volume,  is  never  strictly  true,  even 
for  gases,  and  for  saturated  vapors  it  is  quite  approximate.  In 
solids  and  cold  liquids  the  vapor-tension  becomes  an  insignificant 
affair.  While  neither  pressure  nor  volume  is  ever  exactly  pro- 
portional to  the  temperature,  the  very  fact  that  the  departure 
from  proportionality  is  least  and  almost  zero  in  the  gases,  and 
is  greatest  in  the  solids,  is  itself  a  guide  to  the  understanding 
of  temperature — as  will  be  developed  later. 

Entropy.  The  prime  characteristic  of  entropy  is  its  indiffer- 
ence to  reversible  or  elastic  work-performance,  by  or  upon  the 
body,  with  accompanying  temperature-change.  Its  secondary 
characteristic  is  its  sensitiveness  to  and  inseparability  from 
impact,  friction,  conduction  and  radiation.  Its  third  charac- 
teristic is  its  general  proportionality  with  volume.  While  the 
relations  between  entropy  and  volume  are  even  less  regular  than 
those  between  pressure  or  volume  and  temperature,  yet  they 
are  obvious.  They  stand  out  most  plainly  in  the  process  of 
vaporization,  wherein  the  increase  in  both  volume  and  entropy  is 
large,  and  is  exactly  proportional. 


TEMPERATURE   AND   ENTROPY  143 

The  last  characteristic  of  entropy  to  be  mentioned,  but  not 
the  least  in  importance,  is  its  general  proportionality  to  elas- 
ticity. This,  while  inexact,  like  volume,  is  yet  seldom  reversed. 
An  increase  in  entropy  almost  always  means  an  increase  in 
elasticity.  It  is  this  fact  which  maintains  thermodynamic  hap- 
penings in  an  equilibrium  which  is  almost  always  stable.  Impact 
and  friction  are  features  of  inelasticity.  Impact  and  friction 
always  develop  entropy.  Entropy  is  always  accompanied  by 
elasticity.  Elasticity  reduces  impact  and  friction.  Thus,  the 
phenomena  which  are  due  to  a  deficit  of  elasticity  result  in  the 
cancellation  of  that  deficit  and  a  retardation  of  the  phenomena. 

Labority  and  Thermogy.  The  facts  needed  to  complete 
the  data  for  the  mechanical  definition  of  temperature  and 
entropy  are  those  of  the  two  basic  thermodynamic  processes 
already  defined  in  the  preceding  paper,  viz :  labority  and  ther- 
mogy.  Labority,  or  work-performance,  is  identified  with  tem- 
perature-change, but  does  not  affect  entropy.  Thermogy  is 
identified  with  entropy-change,  but  does  not  itself  affect  tem- 
perature. 

Radial  and  Tangential  Action.  It  is  next  to  be  noted  that 
work-performance  by  heat  is  always  accomplished  by  an  elastic 
movement  of  the  surface  of  the  hot  body  normally  to  itself ;  that 
is  to  say,  radially  in  reference  to  the  molecules  composing  the 
surface-layer.  And  if  the  expansion  of  the  body  is  supposed  to 
be  effected  by  a  simultaneous  expansion  of  all  of  its  molecules, 
this  too  would  obviously  be  a  radial  movement  on  the  part  of 
each  one. 

Thermogy,  on  the  other  hand,  consists  of  an  inelastic  move- 
ment of  the  surface  of  the  hot  body  in  its  own  plane ;  that  is  to 
say,  tangentially  in  reference  to  the  molecules  of  the  surface- 
layer.  In  the  case  of  rubbing  friction,  this  fact  is  obvious.  In 
the  case  of  direct  impact  it  is  less  obvious.  But  even  here  it  is 
only  necessary  to  note  that  impact  consists,  by  its  very  definition, 
of  the  energy  which  is  not  returned  radially  or  elastically,  to  see 
that  impactive  energy  must  also  be  tangential '  in  its  mode  of 
transfer. 

As  to  conduction  and  radiation,  all  that  can  be  said  is  that 
nothing  positive  is  known  as  to  the  form  of  their  action,  but 
that  their  results  are  identical  with  those  of  impact  and  friction. 
Until  positive  evidence  to  the  contrary  arises,  therefore,  it  is 


144  ENERGY 

natural  to  assume  that  impact,  friction,  conduction,  radiation,  and 
even  electrical  resistance,  are  all  identical,  as  to  their  form  of 
molecular  transfer ;  that  is  to  say,  that  they  are  all  tangential,  and 
not  radial,  activities. 

It  is  now  to  be  recalled  that  in  the  study  of  the  elementary 
mechanical  mass-pair  there  was  found  a  marked  contrast  between 
the  radial  and  tangential  funds  of  purely  mechanical  energy. 
The  radial  fund  was  easily  defined  with  mathematical  accu- 
racy. It  was  the  perceptible  fund  of  energy.  It  constituted  the 
medium  of  communication  with  outside  mass-systems.  Its 
intensity  was  what  determined  whether  the  energy  should  undergo 
transformation  or  not :  kinetic  intensity  above  the  critical  point 
producing  dissociation;  spacial  intensity  (of  concentration,  or 

propinquity,  or—)  above  the  critical  point  producing  collision. 

k0 
The  tangential  energy-fund  of  the  pair,  on  the  other  hand, 

was  found  to  be  incapable  of  exact  definition,  although  alterations 
in  it  could  be  measured  exactly.  It  was  the  imperceptible  or 
latent  fund  of  energy.  It  was  self-contained  within  the  mass- 
pair.  It  carried  on  no  direct  communication  with  outside  sys- 
tems. Nevertheless,  it  was  capable  of  receiving  or  sending  ener- 
getic messages  to  the  outside  world,  through  the  medium  of  the 
radial  energy,  to  an  indefinite  extent. 

Mechanical  Concepts  of  Temperature,  Entropy  and 
Heat.  The  data  and  mechanism  for  the  comprehension  of  ther- 
modynamic  happenings,  as  activities  of  mass,  motion  and  space, 
are  now  before  the  eye,  fairly  complete. 

Heat  consists  in  both  motion  and  space  between  the  molecules 
of  the  hot  body.  Each  molecule  is  itself  a  complex  system  of 
particles,  possessing  motion  and  space  relatively  to  each  other; 
and  these  internal  relationships  are  also  heat,  in  part  at  least.  We 
are  not  attempting  to  define  the  molecule  itself,  nor  the  atoms 
which  form  parts  of  a  molecule,  nor  the  ions  or  electrons  which 
may  form  parts  of  the  atoms.  That  is  for  the  chemists  and 
physicists  to  do.  All  that  is  being  said  here  is  that,  if  heat  is  to 
be  regarded  as  a  mode  of  mechanical  motion  at  all,  it  must  be 
regarded  as  a  very  complex  system  of  motions  and  spacial 
separations,  some  of  them  between  the  molecules,  and  some  of 
them  within  the  molecules.  In  the  gases  the  motion  is  chiefly 
between  the  molecules,  each  molecule  moving  bodily  along  hyper- 


TEMPERATURE   AND   ENTROPY  145 

bolic  paths  which  are  almost  straight  lines ;  but  some  of  the  heat- 
energy  still  remains  within  each  molecule.  In  the  solids  the 
motion  is  chiefly  within  each  molecule,  but  there  is  still  some 
motion  between  each  two  molecules,  of  tiny  projectile  particles 
if  not  of  the  whole  molecule. 

These  relative  motions  are  of  two  sorts,  or  components,  viz: 
radial  and  tangential.  Each  orbit  contain?  some  of  each.  The 
elliptical  orbits  have  more  tangential  component  than  radial,  the 
hyperbolic  orbits  more  radial  than  tangential. 

Each  molecule  contains  a  nucleus  of  particles  moving  in 
elliptic  orbit,  and  a  swarm  of  satellites  moving  in  hyperbolic 
orbit.  In  the  solids  the  nucleus  is  the  major  feature,  as  to  mass, 
and  the  satellites  are  the  minor;  in  the  gases  the  satellites  are 
the  major  and  the  nucleus  the  minor.  In  fact,  vaporization  may 
consist  in  the  nucleus  becoming  satellitic.  In  the  so-called  "per- 
fect," or  absolute,,  gas  the  entire  mass  would  be  satellitic.  In  the 
absolute  solid  the  entire  mass  would  be  nuclear.  Neither  condi- 
tion is  ever  attained  in  nature. 

Volume.  Each  tiny  mass-pair  embodies  a  certain  degree  of 
spacial  or  radial  separation,  which  vibrates  constantly  above  and 
below  its  "mean  energetic  distance."  In  the  mean  energetic  con- 
dition the  gravitational  or  centripetal  attraction  is  balanced 
against  the  centrifugal  force  of  tangential  motion.  The  mean 
energetic  distance  of  separation  at  which  this  equilibrium  is  found 
determines  the  volume  of  the  hot  body. 

In  the  solids  the  mean  energetic  distances  are  very  small  and 
the  tangential  velocities  very  high.  In  the  gases  the  distances  are 
very  large  (comparatively  speaking)  and  the  tangential  velocities 
low.  For  at  larger  radii  smaller  tangential  velocities  are  sufficient 
to  maintain  equilibrium. 

The  radial  velocities,  on  the  other  hand,  vary  in  the  reverse 
order.  In  the  solids  they  are  low  and  in  the  gases  they  are  high. 

Temperature  and  Pressure:  In  this  intricate  swarm  of 
particles  of  all  sizes,  moving  in  orbits  of  all  dimensions,  forms 
and  velocities,  the  net  or  integrated  effect  of  all  the  radial  activi- 
ties is  both  temperature  and  pressure.  The  integrated  intensity 
of  energy  is  temperature.  The  integrated  momentum  is  pressure. 

Matter,  even  in  its  most  quiescent  states,  finds  itself  always 
in  contact  with  other  matter,  in  solid,  liquid  or  gaseous  condi- 
tion. At  the  point  of  contact  both  temperature  and  pressure  are 


146  ENERGY 

exerted.  If  a  temperature-difference  exists  at  the  point  of  con- 
tact, energy  is  transmitted  across  the  gap  between  the  bodies,  in 
the  form  of  heat;  but  the  gap  does  not  move.  If  a  pressure- 
difference  exists  at  the  point  of  contact,  no  energy  is  transmitted 
across  the  gap ;  but  the  gap  itself  moves,  and  energy  is  imparted 
in  the  form  of  work. 

In  the  first  case  the  energy  transmitted  comes  from  the  in- 
ternal, or  molecular,  energy  of  the  matter  immediately  adjacent 
to  the  point  of  contact.  In  the  second  case  the  energy  trans- 
mitted comes  either  very  slightly  or  not  at  all  from  this  con- 
tiguous matter.  In  the  case  of  "transient"  energy  it  comes  from 
some  more  or  less  distant  point,  and  reaches  the  point  of  contact 
solely  as  the  pressure-energy  of  a  mass  which  itself  moves 
bodily.  In  the  case  of  energy  developed  adiabatically  by  the 
expansion  of  the  body  in  contact  itself,  it  originates  all  over  the 
body,  between  each  nucleus  and  its  satellites,  simultaneously ;  and 
is  then  transmitted  "transiently"  to  the  particular  satellites  en- 
gaged in  "contact." 

When  temperature-difference  exists  at  the  point  of  contact, 
the  two  bodies  may  or  may  not  exchange  temperature.  That 
depends  entirely  upon  the  pressures  prevailing.  The  only  thing 
certain  is  that  they  always  exchange  entropy.  A  pure  illustration 
— that  is,  one  where  the  process  is  not  complicated  by  the  influ- 
ence of  pressure-changes — is  the  multiple-effect  evaporator,  in  a 
sugar-works.  There  heat  is  transmitted  from  steam  condensing 
under  a  higher  constant  pressure  and  temperature,  on  one  side 
of  a  metallic  wall,  to  water  evaporating  under  a  lower  constant 
pressure  and  temperature  on  the  other  side.  The  wall  serves  to 
neutralize  the  pressure-difference,  but  performs  no  thermal  office. 
The  two  steam-bodies  exchange  only  entropy.  Neither  under- 
goes temperature-change. 

But  they  can  exchange  entropy  only  when  a  temperature- 
difference  exists.  The  office  of  temperature  is  apparently  what 
might  be  called  catalytic :  to  effect  the  transfer  of  entropy,  with- 
out being  itself  affected. 

Unfortunately,  all  of  our  ideas  concerning  heat  are  so  instinct- 
ively founded  upon  the  sense  of  hotness  to  the  touch,  as  the 
prime  criterion  of  thermal  conduction,  that  this  simple  phenom- 
enon is  hard  to  understand.  We  say  that  a  substance  is  "hot," 
when  we  touch  it,  because  we  find  that  the  finger  has  been 


TEMPERATURE   AND   ENTROPY  147 

heated  by  it.  But  if  the  finger  be  not  heated,  any  amount  of 
heat  might  be  transmitted  to  it,  from  a  substance  of  much 
higher  temperature,  yet  we  should  not  say  that  the  latter  was 
"hot'' — except  by  inference.  Thus,  the  steam-hot  water  in  a 
boiler,  for  instance,  would  not  say,  could  it  speak,  when  a  white- 
hot  oil-flame  were  set  against  the  boiler-shell,  that  the  flame  was 
hot;  for  the  water  would  not  be  scorched  thereby,  nor  even 
raised  in  temperature  at  all;  nor  affected  in  any  way  except  to 
be  changed  into  steam  no  hotter  than  the  water  was  before. 

Similarly,  when  one  touches  a  hot  flat-iron  with  wetted  finger, 
the  same  situation,  with  its  lack  of  sensation  of  heat,  is 
approached.  We  judge  that  the  iron  is  hot,  not  because  it  burns 
the  finger,  but  because  it  makes  a  "siss"  of  steam.  We  feel  no 
hotness.  We  merely  hear  the  rapid  formation  of  entropy -and 
volume. 

Now  this  little  experiment  gives  the  only  true  idea  of  thermal 
conduction,  not  as  a  thing  which  reveals  temperature,  but  as  a 
thing  transmitting  entropy — which  last  we  cannot  perceive  at  all, 
except  by  its  volumetric  effects  in  steam-making.  The  fact  is 
most  difficult  to  grasp,  because  all  our  lives  we  have  grown  accus- 
tomed to  telling  whether  things  were  hot  or  cold  by  trying  if  the 
finger  were  heated  or  cooled  in  touching  them.  Nor  have  our 
physical  laboratories,  where  Lord  Kelvin's  ideas  are  well  known, 
done  all  which  it  seems  they  might  to  teach  their  students  better. 

Yet  the  truth  of  the  matter  is  that  the  only  situation  where 
temperature  can  really  be  felt  is  that,  for  instance,  of  the  steam- 
engine  piston.  This  piston  might,  and  ought  to,  be  of  the  same 
temperature  as  the  steam  pressing  against  it.  There  would  then 
be  no  transfer  of  heat  between  them.  The  piston  would  not  be 
heated.  Yet  here  alone  temperature  could  be  felt.  The  piston, 
because  of  the  superior  temperature  of  the  expansive  steam  (as 
compared  with  that  of  the  same  substance  on  its  opposite  face), 
might  be  moved,  could  it  speak,  to  say :  "I  am  propelled;  there- 
fore I  am  impelled  to  note  that  this  steam  on  one  side  of  me  is 
hot!"  For,  if  Carnot  and  Kelvin  knew  anything  about  the 
matter,  it  is  work-performance  alone,  and  not  heat-transfer, 
which  is  the  only  true  sign  of  the  presence  of  temperature. 

With  our  ideas  thus  clarified  it  may  be  accepted  that  the 
transfer  of  heat,  by  the  tiny  radiating  satellitic  particles  of  a  hot 
body,  need  not  be  associated  at  all  with  the  development  of  tern- 


148  ENERGY 

perature  in  the  body  affected.  Therefore  the  puzzle  as  to  the 
mechanical  explanation  of  pressure,  temperature  and  entropy 
resolves  itself  into  the  following  simple  array  of  conditions : 

1.  Temperature  has  already  been  identified,  in  the  case  of 
gaseous  matter,  by  the  mechanical  theory  of  heat,  as  the  kinetic 
energy  of  the  active  particles. 

2.  Temperature   is   manifested   solely   by   elastic   work-per- 
formance; and  work-performance  always  occurs  normally  to  the 
surface  of  the  body,  or  radially  as  to  the  molecules. 

3.  Heat-transfer,  by  thermal  conduction,  has  proven  to  be 
identical  in  its  effects  with  impact  and  friction;  and  impact  and 
friction  have  already  been  identified  with  ^elasticity,  or  molecular 
action  tangential  to  the  molecule. 

4.-  Heat-transfer  takes  place  without  motion  of  the  envelop 
demarking  the  bodies  between  which  conduction  occurs ;  and  the 
only  way  in  which  satellites  flying  along  conic-section  orbits 
might  transfer  energy  from  one  swarm  to  another,  without 
motion  of  the  surface  bounding  the  swarm  normally  to  itself,  is 
tangentially. 

5.  Pressure,  however,  which  is  accompanied  by  no  transfer 
of  energy  across  the  bounding  surfaces,  might  be  exerted  radially 
by  the  flying  particles,  under  the  conditions  stated. 

It  is  from  this  basis  that  emanated  the  statement,  given  above, 
that  both  temperature  and  pressure  are  manifested  by  the  radial 
component  only  of  the  motion  of  the  satellitic  particles.  Tem- 
perature is  their  integrated  intensity  of  energy.  Pressure  is  their 
integrated  momentum.  Temperature-heat  is  the  radial  compo- 
nent only  of  the  radial  energy,  which  was  diefined  for  the 
elementary  mass-system  on  page  40. 

Thermogy.  But,  further,  it  can  be  added  that  thermal  con- 
duction, like  impact  and  friction,  is  the  tangential  transfer  of 
energy  from  the  satellites  of  one  swarm  to  those  of  another. 
This  can  be  accomplished  only  by  means  of  the  tangential  motion 
remaining  at  or  near  the  apastron  tip  of  the  orbit.  Even  when  a 
solid  steel  boiler-shell,  for  instance,  the  molecules  of  which  must 
embody  chiefly  tangential  motion,  is  heated  by  a  white-hot  gas, 
the  molecules  of  which  must  embody  mostly  radial  motion,  the 
energy  transferred  must  be  regarded  as  only  the  tangential  com- 
ponent of  the  gaseous  particles.  Because  the  latter  possess  but 
slight  tangential  motion  (although  plenty  of  radial  energy)  there 


TEMPERATURE   AND   ENTROPY  149 

prevails  but  a  low  rate  of  heat-transfer.  The  highly  eccentric 
orbits  of  the  gaseous  particles,  beautifully  adapted  for  elastic 
work-performance,  are  very  poorly  adapted  for  thermal  con- 
duction. Low  rates  of  heat-transfer,  per  unit  of  surface  and 
difference  of  temperature,  are  broadly  characteristic  of  gaseous 
substances.  Neither  the  gases,  the  satellites  of  which  are  many, 
but  of  too  highly  eccentric  orbit,  nor  the  solids,  the  satellites  in 
which  possess  chiefly  tangential  motion,  but  are  too  few  in  num- 
ber and  small  in  mass,  transmit  heat  well  by  contact.  It  is  the 
liquids,  the  satellites  in  which'  present  the  most  powerful  com- 
bination of  both  numbers  or  mass  (entropy)  and  tangentiality  of 
motion,  which  are  the  best  thermal  conductors. 

But  this  tangential  transfer  of  energy  at  the  apastron  tip  of 
the  satellitic  orbit  cannot  occur  unless  there  exists  a  difference  of 
temperature,  or  radial  intensity,  between  the  two  swarms.  Two 
bodies  manifesting  the  same  temperature,  as  well  as  the  same 
pressure,  must  have  equality  in  both  mass  and  velocity  of  satellites. 
But  if  the  temperature  of  body  A  be  higher  than  that  of  B,  while 
their  pressures  remain  equal  (that  is,  if  momentums  remain  equal 
while  kinetic  energies  are  higher  in  A),  then  the  mass  of  A's 
satellites  must  be  smaller  and  their  velocity  higher  than  in  B. 
But  the  satellites  of  each  swarm  approach  the  other  swarm  in 
all  directions,  chiefly  oblique.  Radially  their  directions  and 
momentums  are  opposed,  and  neutralize  each  other.  But  tan- 
gentially  many  of  them  may  coincide.  In  that  case  the  particle 
of  more  rapid  motion,  as  of  A,  will  overtake  tangentially  and 
impart  energy  to  the  more  slowly  moving  particle  of  the  colder 
body  B. 

Labority.  Elastic  work-performance  by  temperature-heat, 
on  the  other  hand,  is  just  what  these  white-hot  gases  are  best 
fitted  for ;  for  that  is  a  purely  radial  action.  This  fact  is  seen 
in  the  therrnodynamic  superiority  of  the  gas-engine  over  the 
steam-engine.  When  a  body  performs  work  by  heat,  the  radially 
flying  particles  find  themselves  exerting  pressure  against  (that  is, 
revolving  about,  at  the  remote  end  of  their  orbits)  the  satellites 
of  molecular  nuclei  which  are  retreating.  They  are  like  tennis- 
balls  thrown  at  the  rear  end  of  a  retreating  freight-train,. or  a  jet 
of  water  impinging  against  a  retreating  Pelton-wheel  vane.  Their 
direction  of  motion  is  reversed ;  but  they  return  with  their  radial 
velocity  much  reduced. 


150  ENERGY 

Their  tangential  velocity,  however,  remains  unimpaired;  for, 
if  the  process  has  been  a  pure  one,  there  has  been  no  heat- 
transfer.  Their  orbits  are  therefore  reduced  in  eccentricity  and 
increased  in  mean  energetic  distance.  The  molecule  has  lost  its 
temperature  and  pressure,  but  gained  in  volume.  It  has  lost  its 
ability  to  impress  itself  upon  outside  systems,  but  it  has  not  lost 
its  entropy,  or  the  mass-factor  or  quantity-factor  of  its  heat, 
which  gives  heft  to  its  energy.  It  retains  to  the  full  its  tangential 
components.  Its  entropy  remains  constant.  It  is  no  longer 
valuable  for  work-performance,  but  it  is  still  most  efficient  for 
heating-purposes. 

Since  it  will  shortly  be  stated  that  this  quantity-factor  or 
entropy  is  merely  the  number  and  mass  of  the  energy-pairs  at 
work,  it  is  to  be  noted  here  that  there  is  nothing  about  this 
process  of  work-performance  which  should  lead  to  any  con- 
solidation of  mass,  or  diminution  in  the  number  of  mass-pairs 
at  work.  Work-performance,  whether  done  mechanically  or 
thermodynamically,  always  tends  to  occur  at  constancy  of  mass- 
pairing.  Only  interference  by  collision  makes  it  otherwise. 

The  obverse  of  this  process,  adiabatic  compression,  is  not  so 
obviously  explained.  It  is  plain  that  the  approach  of  each  two 
molecules,  enforced  by  outside  power,  must  decrease  their  vol- 
ume. It  is  easy  to  see  that  the  increased  momentum  of  the 
greater  number  of  projectiles  then  met  and  reversed,  per  unit  of 
area,  must  increase  the  pressure.  But  it  is  not  so  clear  that  this 
must  also  increase  the  radial  activity,  or  temperature. 

This  will  be  plain,  however,  when  it  is  remembered  that  the 
forces  which  act  upon  any  satellite  at  periastron  are  the  result, 
not  only  of  mass,  but  of  propinquity.  When  the  molecules  are 
compressed  each  satellite  finds  itself  forced  into  greater  simulta- 
neous propinquity,  at  periastron,  with  a  greater  number  of  nuclear 
masses.  Its  periastron  velocity  therefore  increases ;  and  this 
velocity,  although  tangential  in  direction,  has  already  been 
pointed  out  as  being  largely  true  radial  energy. 

There  also  occurs  the  direct  effect  upon  the  radial  velocity  of 
the  approach  of  the  external  masses  about  which  the  satellites 
are  reversed,  just  as  in  expansion.  Owing  to  the  approach  of 
these  masses,  like  a  freight-train  or  Pelton  wheel  being  backed 
up  against  the  projectiles  which  strike  it,  these  are  returned  with 
increased  radial  velocities. 


TEMPERATURE   AND   ENTROPY  151 

Temperature.  Temperature,  then,  is  radial  kinetic  energy. 
Work-performance  by  temperature-drop  is  in  no  way  different 
from  the  performance  of  work  by  the  velocity-reduction  of 
mechanical  mass,  as  in  a  turbine  or  Pelton  water-wheel.  Tem- 
perature-increase by  compression  is  in  no  way  different  from  the 
acceleration  of  any  mass  by  a  supply  of  energy,  as  in  the  cen- 
trifugal pump  or  marine  propeller.  The  engineer  who  can  under- 
stand these  mechanical  devices  can  understand  adiabatic  thermo- 
dynamics. 

The  doctrine  that  temperature  is  kinetic  energy  is  not  new,  at 
all.  What  these  papers  are  intended  to  emphasize,  in  connection 
with  it,  is  that  the  kinetic  energy  which  constitutes  temperature  is 

1.  The  radial  component  only  of  the  molecular  energy; 

2.  It  is  embodied  in  mass-particles  very  much  smaller  than 
the  entire  molecule,  as  well  as  in  the  latter  itself ; 

3.  These  facts  apply  to  liquids  and  solids  as  well  as  to  gases ; 

4.  The  "rebound"  of  the  flying  particles  cannot  be  imagined 
as  that  of  perfect  elasticity  in  collision,  but  must  be  accepted  as 
consisting  of  circumrevolution  without  contact. 

Entropy.  Turning  now  to  the  question  of  the  mechanical 
nature  of  entropy,  and  of  its  generative  process,  thermogy,  it 
might  be  inferred  that  these  were  to  be  found,  because  of  their 
contrast  with  temperature  and  labority,  in  the  tangential  motion 
of  the  particles.  They  are,  indeed,  closely  connected  therewith. 
Thermogy  is  the  acceleration  of  the  particles  in  their  tangential 
component.  But  entropy  is  the  result  of  this  process,  not  the 
process  itself ;  and  the  natural  result  of  an  increase  in  tangential 
energies  is  a  separation  or  scattering  or  subdivision  of  the  total 
mass  into  a  greater  number  of  mass-pairs  capable  of  embodying 
temperature. 

Entropy  is  what  corresponds  to  the  quantity  factor  MXM2  of 
the  elementary  mechanical  mass-pair.  It  is,  more  accurately,  the 
£  (MM)  of  any  mechanically  energetic  system.  It  is  the  X  of 
Fig.  7,  a  function  of  n  the  number  of  portions  into  which  the 
total  mass  is  subdivided.  It  is  the  quantity-factor  N  of  the 
thermal  diagram,  Fig.  8.  It  is  the  degree  of  subdivision  and 
dispersion  of  the  nuclei  A  and  B  of  Fig.  9  into  satellitic  swarms. 
It  is,  in  short,  the  degree  of  subdivision,  specialization  and 
organization  of  an  originally  comparatively  unified,  solid,  rigid, 
inert  and  inelastic  mass,  into  a  system  capable  of  embodying  that 


152  ENERGY 

% 

space  and  motion  which  are  to  us  the  sole  visible  manifestations 
of  energy  and  elasticity.  This  quantity-factor  is  always  itself 
imperceptible;  that  is  to  say,  directly  imperceptible;  but  it  is 
what  gives  "heft"  and  power  to  that  which  alone  is  directly 
perceptible,  viz  :  space  and  radial  motion. 

Intramolecular  Equilibrium.  It  is  the  stability  of  equilib- 
rium within  the  molecule  which  necessitates  always  a  readjust- 
ment of  either  sort  of  energy,  radial  or  tangential,  whenever  any 
alteration  is  made  in  the  other.  Thus,  in  heating  ice  or  water 
isomorphically,  only  thermogy  is  performed  directly.  The  altera- 
tions of  entropy  occur  at  constant  temperature;  the  subdivision 
of  the  molecules  results  from  speeding  them  up  tangentially,  as 
a  potter  does  his  wheel,  and  not  from  any  radial  acceleration  or 
rise  in  temperature.  But  the  immediate  indirect  effect  of  this 
thermogy,  in  the  face  of  a  superior  external  pressure  upon  the 
molecule,  is  to  upset  its  internal  equilibrium;  so  that  the  energy 
which  was  imparted  tangentially,  expanding  the  molecule  as  a 
fly-ball  governor  expands  by  acceleration,  is  squeezed  out  radially, 
by  the  recompression  of  the  molecule  adiabatically  by  the  external 
pressure,  into  an  increase  in  radial  components  which  we  call 
temperature-rise.  The  ice  or  water  gets  warmer  as  it  is  pounded 
or  heated;  but  this  occurs  only  because  (i)  it  has  first  been  so 
subdivided,  isothermally,  by  the  thermogy,  as  to  have  become  more 
elastic  and  susceptible  of  compression;  and  (2)  because  the 
surplus  pressure  requisite  for  the  recompression  of  this  greater 
elasticity  is  present.  If  the  surplus  pressure  be  not  present  the 
temperature  will  not  rise.  It  is  the  adiabatic  recompression  of 
the  elastic  portion  of  the  molecule,  by  this  apparently  static 
external  pressure,  and  not  the  thermogic  addition  of  energy  to 
its  inelastic  portion  by  pounding  or  heating,  which  raises  its 
temperature. 

On  the  other  hand,  when  thermogic  impact  or  friction  or 
heat-conduction  leads  to  the  melting  of  the  ice  or  the  vaporiza- 
tion of  the  water,  without  incidental  rise  in  temperature,  it  is 
because  the  accumulation  of  internal  radial  intensity  has  become 
sufficient  to  counterbalance  the  static  external  pressure.  The 
equilibrium  has  become  indifferent.  The  expansion  of  the  mole- 
cule by  thermogic  increase  of  its  tangential  velocity  now  jumps  it 
into  a  new  condition  of  stable  equilibrium,  that  of  saturated  steam 
—like  a  fly-ball  governor  breaking  some  of  its  resisting  links— 


TEMPERATURE  AND  ENTROPY  153 

and  in  that  condition  it  remains  stably.  No  reactive  compression 
can  occur  to  raise  the  temperature,  and  we  therefore  say  that 
vaporization  occurs  isothermally. 

In  any  such  a  thermogic  action  the  chief  absorbent  of  energy 
would  be  the  separation,  or  disgregation,  of  the  particles.  It  was 
pointed  out  in  Chapter  I  that  the  expression  for  space-energy 

was  proportional  to  -^ ~-,  and  that  in  this  expression  the  value 

b0       b 

of  S0  had  much  more  to  do  with  the  value  of  the  function  than 
did  that  of  S.  In  other  words,  when  mass  is  separated  the  work 
involved  depends  much  more  upon  the  original  degree  of  con- 
centration, or  density,  of  the  mass  than  upon  the  final  degree  of 
diffusion.  This  work  of  separation  is  called  disgregation-work 
— the  word  disgregation  signifying  the  "scattering  of  a  flock,"  or 
the  opposite  of  congregation.  This  disgregation  work  is  very 
great  for  solids  and  liquids,  because  of  their  comparatively  great 
density.  They  are  unusual  congregations  or  concentrations  of 
mass,  embodying  unusual  deficits  of  energy.  For  gases  it  is  much 
less,  decreasing  to  almost  zero  as  the  diffusion  becomes  great. 
But  because  mass  can  never  be  so  widely  separated  that  its 
mutually  attractive  force  becomes  zero,  so  no  gas  can  ever 
become  so  rarefied  or  "perfect"  that  its  disgregation-work  in 
change  of  volume  ever  becomes  zero. 

In  illustration  of  this  mechanical  concept  of  the  thermal  mole- 
cule, the  writer  has  used  the  simile  of  a  juggler  standing  before 
a  table  carrying  a  stock  of  balls.  The  total  stock  of  balls  repre- 
sents the  molecule.  The  portion  which  the  juggler  succeeds  in 
keeping  in  motion  in  the  air  is  the  satellitic  portion.  The 
remainder  upon  the  table  is  the  nucleus.  In  this  simile  the 
nucleus  would  have  no  motion  at  all,  whereas  in  the  actual 
molecule  the  nucleus  possesses  plenty  of  motion,  only  reduced 
to  an  eccentricity  below  unity.  In  the  simile  the  "satellites" 
possess  merely  elliptic  motion,  whereas  in  the  molecule  they  pos- 
sess hyperbolic  motion.  But  still  the  simile  may  help  the  under- 
standing; and  in  it  the  juggler's  energy  of  action  must  be  sup- 
posed to  be  the  energy  inherent  in  the  balls  themselves. 

When  the  juggler  exhibits  little  energy  only  a  small  portion 
of  the  balls  will  be  kept  in  the  air  at  once.  The  bulk  of  the  stock 
of  mass  on  hand  remains*  on  the  table ;  that  is,  in  the  central 
nucleus  of  the  molecule.  As  the  juggler  gains  energy,  however, 


154  ENERGY 

the  flying  balls  will  remain  longer  in  the  air,  and  the  idle  stock 
on  hand  must  be  drawn  upon  in  order  to  supply  fresh  recruits. 
In  this  simile,  the  total  energy  of  the  flying  balls  (including  their 
potential  energy)  is  the  heat  of  the  molecule;  their  vertical 
kinetic  energy  per  unit  of  mass  is  its  temperature;  the  distance 
they  fly  is  its  volume;  the  force  with  which  they  might  impinge 
upon  a  ceiling  overhead  is  its  pressure ;  and  the  degree  to  which 
the  total  stock  of  balls  is  subdivided  into  possible  pairs  by  the 
juggler's  activity  (calling  the  idle  stock  on  the  table  a  solidified 
unit)  is  its  entropy. 

The  following  picture  of  the  physical  nature  of  entropy  is 
taken  from  the  author's  article  in  the  November  (1908)  issue  of 
the  Harvard  Engineering  Journal : 

"Every  preparatory  student  understands  entropy  thoroughly, 
although  he  has  never  been  taught  to  know  it  by  name.  For 
every  school-boy  knows  well  the  value  of  organization  for  team- 
work upon  the  athletic  field.  He  knows  that,  for  good,  effective 
sport,  a  given  mass  of  men  must  be  used,  not  as  a  single  massive 
unit,  nor  yet  as  a  number  of  independent  individuals;  but  by 
subdivision,  specialization  and  organization  it  must  be  animated 
into  an  organic  whole. 

"Indeed,  without  subdivision  into  at  least  two  sides  there  can 
be  no  game  at  all ;  just  as  in  mechanical  energy  there  can  be  no 
energy  until  the  mass  present  is  subdivided  into  at  least  two 
portions,  which  mutually  oppose  and  react  upon  each  other. 
Similarly,  the  particular  sort  of  energy  called  athletic  antagonism 
consists  in  the  reaction  occurring  'between  two  opposed  sides.  It 
does  not  lie  in  either  side  alone,  nor  yet  in  the  individuals  alone — 
though  each  of  the  latter  possesses  his  own  fund  of  internal 
energy  (due  to  his  subdivision  into  a  variety  of  organs,  muscles, 
glands,  etc.).  For  in  a  company  of  even  the  most  stalwart 
athletes  there  could  be  embodied  no  other  form  of  energy  than 
muscular,  glandular,  etc.,  exhibited  in  individual  feats,  unless 
there  existed  athletic  organization  into  at  least  two  contending 
parties. 

"For  in  all  the  better  sorts  of  athletic  contests,  such  as  base- 
ball and  foot-ball,  there  arises  a  quite  distinct  form  of  energy 
from  the  organic  energy  of  the  individual.  The  Subdivision, 
specialization,  organization  and  interaction  between  players 
applies  not  merely  to  the  mass  as  a  whole,  but  pervades  each  of 


TEMPERATURE   AND   ENTROPY  155 

the  two  'sides/  One  man  trains  to  be  a  pitcher,  another  to  be 
catcher,  and  a  third  to  be  short-stop.  Each  'side'  becomes  itself 
an  organism,  containing  a  form  of  energy  which  is  quite  distinct 
from  the  aggregate  energy  of  the  individual  players.  The  two 
sides  cooperate  competitively  to  the  production  of  a  game.  The 
men  on  each  side  compete  cooperatively  to  the  advancement  of 
that  particular  side  of  the  game.  There  thus  arises  a  more  intri- 
cate and  effective  form  of  contest  than  is  possibly  to  be  had  from 
any  number  of  athletes  whatever,  trained  to  any  degree  of  skill 
and  strength  whatever,  if  each  acts  only  as  an  individual.  There 
has  arisen  a  new  'form'  of  energy:  team-work. 

"This  subdivision  and  specialization  into  team-work  consti- 
tutes athletic  entropy.  Correspondingly,  athletic  temperature  is 
the  intensity  and  vigor  of  play  of  each  individual  player.  Both 
are  cultivated  by  the  coaches,  as  essential  to  good  results;  but 
they  aid  in  those  results  in  distinctly  different  ways.  These  ways 
might  be  contrasted,  for  instance — if  hearsay  be  accepted  as  true, 
for  the  purpose — by  saying  that  Harvard  wins  her  victories  by 
athletic  temperature,  by  brilliant  individual  play,  whereas  Yale 
wins  hers  by  athletic  entropy,  by  a  dogged  devotion  to  team-work 
which  overbears  in  the  long  run. 

''Or  a  similar  contrast  might  be  borrowed  from  the  field  of 
heat-engines.  The  gas-engine,  with  its  tall  and  white-hot  but 
slightly  entropied  cycle,  represents  the  acme  of  efficiency  in 
power-development  from  heat.  Each  molecule  of  its  working- 
substance  moves  at  tremendous  radial  speed,  or  temperature. 
But  the  gas-engine  has  never  been  able  to  compete  success- 
fully, in  the  world's  work,  with  the  steam-engine;  for  the 
latter's  squat,  cool  and  slow-moving  cycle  is  so  heavily  entropied 
that  the  engine  stays  in  the  ring  and  continues  to  push 
hard,  long  after  the  more  delicate  gas-engine  has  'laid  down  on 
the  job.'  " 

This  position  is  to  be  emphasized  more  as  time  passes.  The 
importance  of  team-work  is  ever  upon  the  increase.  As  popula- 
tions increase  inventions  multiply.  A  greater  number  of  people 
are  now  in  daily  communication  with  each  other — and  therefore 
forced  to  cooperate,  whether  they  will  or  no — than  a  century 
ago  could  be  reached  in  a  month's  travel.  Life,  in  shop,  market 
and  legislature,  grows  daily  more  multiplex  and  intricate.  The 
need  for  the  better  and  more  patriotic  understanding  of  human 


156  ENERGY 

relationships  grows  daily  more  urgent,  while  that  for  brilliant 
patriotism  of  individual  deeds  is  on  the  wane. 

All  this  is  inextricably  tied  up  with  this  question  of  entropy, 
or  mass-pairing,  or  the  quantity-factor  of  energy.  In  such  a 
book  as  this  no  more  than  a  suggestion  of  its  broad  significance 
and  importance  can  be  made.  But  to  pass  the  occasion  without 
even  that  would  be  a  grave  mistake.  Our  best  records  in  Ameri- 
can history  are  the  victories  won  by  national  team-work.  In 
1864  it  was  the  solidarity  of  the  North,  in  an  organized  effort  of 
each  for  all,  which  won  over  the  more  intense  valor  and  superior 
tactics  of  the  South,  which  lacked  it.  In  1898  all  the  credit  we 
won  was  due  to  national  entropy,  and  all  our  shame  to  a  lack 
of  it — to  the  false  idea  that  intense  and  spectacular  Rough 
Riding  could  accomplish  what  was  desired,  without  the  coopera- 
tion of  every  individual  in  the  nation  to  the  army's  proper  feeding 
and  nursing. 

The  conclusions  stated  in  the  above  pages  are  of  fundamental 
importance.  They  may  be  summarized  as  follows : 

1.  Temperature  is  the  intensity-factor  of  heat.     It  is  the 
radial  kinetic  intensity  of  the  mass-particles,  large  and  small,  of 

velocity-squared 

the  molecule.     It  is  proportional  to  — — = 7 « r~>  or 

**  total  mass  of  molecule 

mathematically,  to  S(-^).   It  is  affected  solely  by  changes  of 

volume  under  pressure,  contributing  or  abstracting  radial  motion 
only  (or  eccentricity  of  orbit)  to  the  particles  of  the  molecule. 

2.  Entropy   is   the   extensity-factor,   or   quantity-factor,   of 
heat.    It  is  the  extent  of  mass-pairing,  or  degree  of  subdivision, 
of   the   molecule,   into   separate   particles    which   interact   ener- 
getically.   It  is  the  variable  proportion  of  the  mass  of  each  mole- 
cule effective  in  heat-motion.    Mathematically,  it  is  S(MM). 

3.  Labority  is  the  isentropic  alteration  of  temperature,  by 
elastic,  or  radial,  work-performance  or  absorption.    It  is  a  varia- 
tion of  the  intensity  of  thermal  energy,  or  velocity  of  radial 
motion,  under  constancy  of  extensity,  or  degree  of  mass-pairing, 
or  entropy. 

4.  Thermogy   is   the   isothermal   alteration   of   entropy,   by 
inelastic,  or  tangential,  work  or  heat  absorption.    It  is  a  variation  of 
the  extensity  of  thermal  energy,  or  entropy,  or  subdivision  of  the 
molecule,  under  constancy  of  its  radial  intensity,  or  temperature. 


TEMPERATURE  AND  ENTROPY  157 

5.  Both  of  these  processes  occur  in  apparent  purity  in  nature. 
The  first  occurs  in  adiabatic  expansion  or  compression — finding 
almost  pure  instance  in  the  explosion  of  a  steam-boiler,  where 
lack  of  time  excludes  the  conduction  of   heat   to  or  from  the 
outside.    The  second  finds  instance  in  the  vaporization  of  water 
under  constant  pressure. 

6.  All  other  thermodynamic  processes  than  these  consist  of 
combinations  of  the  two — either  produced  simultaneously,  by  two 
outside  causes  acting  at  once,  or  occurring  seriatim,  one  as  the 
result  of  the  other,  within  the  internal  equilibrium  of  the  mole- 
cule.   Instances  of  the  latter  are  the  isomorphic  heating  of  water, 
where    the    temperature    rises    because    the    entropy    has    been 
increased;  or  the  wire-drawing  of   steam,   where  the   entropy 
increases  because  the  temperature  has  fallen. 

Instances  of  the  simultaneous  operation  of  two  external 
influences  occur  in  the  compression  or  expansion  of  gases  in 
actual  cylinders.  In  compression  the  temperature  is  raised  by 
work-performance  and  the  entropy  decreased  by  cooling,  simulta- 
neously. In  expansion  the  temperature  is  dropped  by  work- 
performance  and  the  entropy  decreased  by  cooling.  Every  pos- 
sible combination  of  thermogy,  positive  or  negative,  with  labority, 
positive  or  negative,  is  known  in  power-house  practise ;  and  none 
of  them  can  be  explained  consistently  without  this  clear  distinc- 
tion between  thermogy  and  labority,  as  independent,  though  some- 
times connected,  mechanical  phenomena. 

As  possibly  of  further  aid  in  understanding  a  difficult  subject, 
the  following  parallel  between  temperature  and  entropy  is  given : 

TEMPERATURE.  ENTROPY. 

In  Molecular  Mechanics: 

Intensity  of  heat-energy,  or  space-  Extensity  of  heat-energy,  or  quan- 

and-motion  factor.  tity-factor  of  heat. 

Radial  activity.  Mass-pairing    involved,    maintained 

by    tangential   activity. 

Degree  of  radial  space  and  motion  Extent  of  subdivision  of  molecule 

between    mass-particles   of   mole-  into  mass-particles  radially  sepa- 

cule.  rated  by  space  or  motion. 

U2sin2  2  c 

Proportional  to  e.  -^  +  ^     =e.-^-.  Proportional  to  2  MiMa. 

In  Relation  to  External  Bodies: 

Action  normal  to  body's  surface.  Action  parallel  to  body's  surface. 

Elastic    work-performance    or    ab-       Inelastic    work-absorption,    or    ab- 
sorption, sorption  or  radiation  of  heat. 
Isentropic  labority.                                     Isothermal  thermogy. 


CHAPTER  XIII. 

THE  ENERGETIC  CYCLE. 

Hitherto  the  discussion  has  turned  chiefly  upon  what  energy  - 
is.  It  next  becomes  of  importance  to  consider  how  it  may  be 
obtained  for  human  consumption.  The  question  is  not  one  as  to 
the  sources  from  which  energy  may  be  derived.  Much  has  been 
written  upon  this  topic;  yet,  except  for  the  need  for  endlessly 
repeating  the  lesson  that  the  waves  of  the  sea  do  not  constitute 
a  practicable  source  of  supply,  nor  even  the  tides  (except  under 
most  unusual  circumstances),  there  is  little  to  be  said  in  this 
connection. 

The  question  of  prime  interest  is  not :  How  may  man  acquire 
energy  ?  It  is,  instead :  How  does  nature  do  it  ? 

On  every  hand,  in  every  natural  phenomenon,  is  seen  an 
endless  chain  of  energy-transformations.  The  one  stock  of 
energy  possessed  by  nature  is  made  available  for  the  most  intri- 
cate and  endless  variety  of  purposes  and  effects,  merely  by 
transformation.  In  the  power-house  one  sees  a  little  of  this : 
The  ownership  of  a  store  of  black,  dense,  heavy,  unchanging 
coal  gives  one  permanence  of  potentiality  for  power.  Upon 
demand,  this  may  be  converted  into  heat  and  light,  in  combus- 
tion. But  the  permanence  is  gone;  trans ferability,  and  also 
evanescence,  have  taken  its  place. 

The  evanescence  of  flame-heat  being  usually  too  great,  a  fair 
degree  of  permanence  is  acquired,  with  portability  retained,  by 
transformation  into  steam-heat.  This,  upon  demand,  becomes 
mechanical  motion.  This,  in  turn,  may  become  electricity.  The 
electricity  may  then  become  chemical  energy  again,  the  form 
from  which  it  started,  in  electrolysis ;  or  light  and  heat  again,  as 
it  was  in  the  furnace ;  or  an  ether-wave  carrying  human  intelli- 
gence across  the  midnight  seas ;  or  mere  motion  and  heat  again. 
Even  in  the  human  laboratory  the  transformations  are  startling 
enough.  But  in  Dame  Nature's  they  are  far  more  so;  they  are 
stupendous,  amazing  and  quite  incomprehensible,  except  in  their 
most  general  aspects. 

158 


THE  ENERGETIC  CYCLE  159 

To  nature,  therefore,  the  task  of  keeping  all  her  vast  and 
intricate  processes  in  motion  is  not  one  of  creation,  or  acquisi- 
tion, of  energy,  from  some  other  source.  It  is  merely  that  of 
keeping  what  she  always  possesses  in  active  transformation  and 
circulation.  Does  nature  wish  to  grow  a  world  of  fresh  green 
verdure,  some  spring,  or  rear  an  entire  human  race  in  some 
twinkling  of  the  universal  eye?  She  does  not  transport  a 
vegetation  or  a  mankind  from  some  other  corner  of  the  universe. 
She  merely  places  between  the  sun  and  that  unknown  space  into 
which  it  has  been  radiating  its  heat,  throughout  unrecorded  time, 
a  planet:  an  earth,  watered  and  aired  and  fitted  for  its  mission. 
The  result  follows.  Without  necromancy,  as  naturally  and  mechan- 
ically as  a  properly  made  motor  starts  when  the  current  is  turned 
on,  a  flood  of  verdure  spreads  over  the  face  of  the  earth  when 
spring  comes,  or  a  fauna  of  pterodactyls  and  eohippi  develops 
into  races  of  horses  or  men  when  the  changes  of  season  become 
sufficiently  favorable. 

It  is  quite  similarly  that  man  places  his  simple  little  con- 
trivances in  the  way  of  the  same  vast  currents  of  energetic  flow 
in  nature,  that  he  may  consummate  his  own  tiny  and  short- 
sighted plans.  He  finds  a  reservoir  of  water  up  amid  the  hills, 
and  sees  its  contents  cascade  toward  the  sea.  In  between  the 
lake  and  sea  he  places  his  water-wheels,  and  derives  power.  He 
finds  a  red-hot  fire,  radiating  heat  into  the  lower  realms  of 
temperature.  In  between  fire  and  refrigerator  he  places  his 
steam-engines,  and  derives  power. 

The  plan  is  a  simple  one  to  look  at.  But  how  does  it  work,  in 
terms  of  the  ideas  as  to  heat  and  work  which  the  preceding  papers 
have  outlined  ? 

Take  the  simplest  case  first:  that  of  the  mill-pond,  the  tail- 
race  and  the  old-fashioned  overshot  water-wheel  between  them. 

The  water  in  the  mill-pond  possesses  energy  because  of  its 
separation  from  the  earth.  It  and  the  earth  constitute  two  mass- 
portions,  which  are  paired  or  opposed  in  a  mutual  reaction 
which  we  call  force,  or  "weight,"  and  which  embodies  energy. 
The  two  were  pulled  apart  originally  by  sun-heat,  a  "supply  of 
energy  from  without/'  and  they  tend  to  reunite  as  they  reship 
their  fund  of  energy  upon  its  next  journey  in  the  world. 

Man,  however,  cannot  utilize  an  entire  mill-pond  of  water  at 
once.  He  therefore  takes  it  a  little  at  a  time.  A  cubic  foot  of 


160  ENERGY 

water,  or  a  ton,  say,  will  be  taken  into  the  upper  buckets  of  a 
water-wheel,  to  do  work.  This  ton  of  water  is  detached  from 
the  natural  supply,  and  introduced  into  the  wheel;  and  in  the 
performance  of  this  process  great  care  is  taken  to  have  no  "free 
fall,  friction  or  impact"  occur,  beyond  what  is  unavoidable. 

This  simple  process  of  detaching  energy-bearing  mass  from 
the  stock  of  it  which  nature  brings  to  our  hands,  and  its  embodi- 
ment into  the  machine  with  which  it  is  designed  to  develop 
power,  is  simplicity  itself.  Yet  it  is  necessary  to  call  attention  to 
it  in  this  special  way,  because  it  typifies  a  process  which  is  an 
essential  part  of  all  energetic  cycles,  and  which  appears  to  be,  in 
thermodynamic  cycles,  one  of  the  most  difficult  of  all  processes 
for  the  student  to  understand. 

When  the  stock  of  energy-bearing  mass  is  once  in  the  wheel, 
it  is  permitted  to  lose  its  vertical  separation  from  the  earth,  by 
falling,  under  resistance  and  control,  to  the  tail-race  level.  It  is 
there  detached  from  the  wheel  and  rejected  into  the  tail-race. 

According  to  the  mathematics  of  engineering,  if  W  be  the 
weight  of  water  taken  into  the  wheel  and  h  be  the  height  of 
mill-pond  above  tail-race,  the  energy  transformed  by  the  fall  is 
Wh.  The  cycle  of  events  might  be  portrayed  by  a  plain  rec- 
tangular diagram,  such  as  Fig.  10,  in  which  the  horizontal 
abscissae  measure  weight  and  the  vertical  ordinates  measure 
height.  If  the  point  D  should  represent  the  weight  and  height 
of  the  empty  buckets  at  the  top  of  the  wheel,  DA  the  weight 
of  the  water  taken  in,  OIj  the  height  of  the  mill-pond  above  the 
sea-level  and  OI2  the  similar  height  of  the  tail-race,  then  the 
work  done  by  the  falling  buckets  would  be  measured  by  the 
rectangle  IjABLj  and  that  by  the  rising  buckets  by  the  rectangle 
^CDL,.  The  net  difference,  or  the  rectangle  DABC,  would 
measure  the  net  work  done,  or  Wh.  The  efficiency  with  which 

the  total  available  fall  had  been  utilized  would  be  p  AX  x  =T)I^' 
But  under  modern  conditions  the  old-fashioned  overshot  or 
direct-gravity  water-wheel  has  had  to  give  place  to  the  turbine. 
For  handling  large  quantities  of  water  under  heads  which  often 
exceed  the  largest  practicable  diameter  for  wheels,  the  turbine 
is  far  superior  to  the  older  form.  For  very  high  heads  and 
small  quantities  of  water,  such  as  occur  in  the  mountains  of  the 
mining-regions,  the  Pelton  wheel  has  been  used  in  preference  to 


THE  ENERGETIC  CYCLE 


161 


the  submerged  turbine.  And  as  the  different  parts  of  the  Pelton 
wheel  are  separated  more  distinctly  to  the  eye,  it  may  form  a 
clearer  illustration  in  the  following  discussion  than  the  turbine. 

For  in  both  of  these  modern  types  of  water-motor  another 
cycle  of  energy-transformations  than  that  just  described  has  been 
interposed  between  the  mill-pond  and  the  tail-race.  The  water 
performs  its  vertical  fall,  not  in  the  moving  machine,  but  in  a 
closed  penstock.  At  the  foot  of  this  penstock  its  energy  has 
been  stored  in  the  form  of  accumulated  pressure.  It  is  then 
chiefly  "transient"  energy,  although  a  slight  portion  is  stored 
elastically  in  the  compression  which  the  water  has  suffered  during 
its  fall. 


'c 


FIG.    IO. 

Before  the  water  is  admitted  to  the  moving  part  of  the  wheel, 
this  transient  and  resilient  energy  is  converted  into  kinetic  energy 
of  jet,  by  the  nozzle  or  guide-blades.  It  is  in  this  form  that  it 
is  received  by  the  wheel  proper,  which  is  so  designed  as  to  be 
adapted  to  the  reduction  of  velocity-relatively-to-the-earth  by  an 
alteration  of  direction  of  velocity-relatively-to-wheel,  by  means 
of  curved  vanes.  The  illustration  of  the  manner  of  doing  this, 
by  means  of  parallelograms  of  velocity,  is  familiar  to  every  tech- 
nical student,  and"  needs  no  reproduction  here. 

The  energetic  cycle  which  is  thus  performed  by  the  nozzle 
and  wheel  in  cooperation  is  quite  similar  to  that  of  the  overshot- 
wheel,  in  principle,  and  may  be  illustrated  equally  well  by  Fig. 
10.  Only,  in  this  new  use  of  Fig.  10  the  vertical  ordinates  must 


162  ENERGY 

be  accepted  as  measuring,  not  height  nor  head  of  static  water- 
energy,  but  the  velocity-squared  of  kinetic  water-energy.  The 
horizontal  abscissae  measure  the  quantity-factor,  usually  taken 
as  mass,  instead  of  weight,  as  before. 

In  this  new  case  it  is  as  true  as  it  was  before  that  the  energy 
taken  into  the  wheel  is  measured  by  the  rectangle  DAXtX2,  that 
rejected  to  the  tail-race  is  BCX,^,  and  that  converted  into  work 

is  DABC.    The  efficiency  of  conversion,  as  before,  is  p^v  X  = 

•=^-  .  The  only  difference  is  that  now  there  are  no  "empty 
if) 

buckets"  to  be  carried  up  and  down,  and  so  the  vertical  axis  OI 
should  coincide  with  CD. 

In  these  two  mechanical  cycles,  which  are  so  familiar  to  all 
engineers  that  their  description  seems  superfluous,  there  are 
visible  all  of  the  characteristics  of  the  most  obscure  thermo- 
dynamic  and  other  energetic  cycles,  if  the  positions  taken  in  the 
preceding  papers  of  this  series  be  true.  It  is  therefore  important 
to  observe  carefully  just  what  has  happened  therein. 

In  the  first  place,  the  question  must  be  reviewed:  What  is 
potential  hydraulic  energy?  What  is  kinetic  hydraulic  energy? 

The  equations  which  were  given  as  exact  answers  to  these 
questions,  in  Chapter  I,  are  these  : 

Potential  energy  =  c  MtM2  (4  ---  5")  (32) 

b0       & 

Kinetic  energy  =-i-M1M2 


In  these  expressions,  it  was  pointed  out,  the  factor  MjM2 
indicated  the  extent  to  which  the  total  mass  present  was  sub- 
divided into  mass-pairs  capable  of  embodying  energy,  and  was 
called  the  extensity  of  energy.  The  remainder  of  each  expression 
indicated  the  degree  of  spacial  or  kinetic  relationship  which  was 
embodied  in  these  mass-pairs,  to  constitute  them  energetic  pairs, 
and  was  called  the  intensity  of  energy. 

The  study  of  the  general  characteristics  of  mass,  space  and 
motion,  as  they  appear  all  about  us  in  nature,  showed  that  each 
of  these  factors  might  be  an  independent  variable.  The  ex- 
tensity  factor,  for  instance,  varies  in  one  direction  by  the  con- 
solidation of  mass,  and  in  the  other  by  its  subdivision.  The 


THE  ENERGETIC  CYCLE  163 

forces  of  gravitational  attraction  are  everywhere  and  all  the  time 
tending  to  produce  consolidation.  The  equally  universal  phe- 
nomena of  motion,  collision,  impact  and  friction,  engendering 
tangential  motion  and  centrifugal  force,  are  always  tending  to 
produce  subdivision  and  comminution  of  matter. 

The  intensity-factors  also  vary  in  either  direction,  and  under 
the  same  forces  or  closely  allied  ones.  Gravitational  attraction  is 
always  tending  to  increase  the  propinquity  between  masses,  and 
likewise  their  velocities.  The  centrifugal  forces  developed  by 
those  velocities  are  always  tending  to  decrease  the  velocities  and 
the  propinquity  simultaneously. 

Two  Basic  Processes  of  Mechanical  Energetics.  The 
two  fundamental  processes  of  mechanics,  therefore,  are : 

1.  The  variation  of  mass-pairing,  or  extensity  of  energy,  by 
the  subdivision  or  consolidation  of  mass,  under  constancy  of 
intensity; 

2.  The  variation  of  velocity  or  propinquity,  or  intensity  of 
energy,  by  the  approach  or  separation,  or  by  the  acceleration  or 
retardation  of  masses,  under  constancy  of  mass-pairing. 

The  second  of  these  processes  is  apparently  the  more  familiar. 
The  performance  of  work  by  falling  weights,  or  by  hammers 
"slowing  up"  against  anvils  or  by  trains  climbing  grades  by 
momentum,  or  by  water  accelerated  in  nozzles  and  retarded 
against  moving  vanes,  are  all  familiar  instances  of  variations  in 
intensity  of  energy  under  constancy  of  mass-pairing. 

But  mixed  in  with  and  confused  with  these  processes  is  the 
first  named  sort,  which  are  distinctly  contrasted  with  them  in 
character.  That  is  to  say,  before  the  weight  can  fall  and  perform 
work  it  must  first  be  released,  or  "detached,"  from  the  earth,  so 
as  to  form  a  mass-pair  reacting  with  it.  The  hammer,  before  it 
can  be  swung  on  the  anvil,  must  first  be  picked  up.  The  water, 
before  it  can  project  against  vane,  must  first  be  detached  from 
earth  and  mill-pond.  The  railroad-train,  before  it  can  climb  a 
grade  by  its  own  motion,  must  first  be  built  and  constituted  a 
thing  separate  from  the  earth :  ore  must  be  mined  and  smelted, 
designs  made,  metal  cast,  wrought  and  machined,  and  the  parts 
assembled  into  a  complex  whole.  The  complete  structure  must 
be  equipped  with  water,  fire,  oil,  etc.  All  this  must  precede  the 
purely  mechanical  process  of  setting  the  train  into  motion  fit  to 
climb  a  grade. 


164  ENERGY 

In  these  examples  it  is  clear  that  great  differences  in  appear- 
ance occur  in  this  one  process  of  subdivision  of  the  originally 
unit  earth  into  a  mass-pair  capable  of  embodying  energy.  In  the 
tripping  of  the  drop  or  the  picking  up  of  the  hammer  it  is  so 
slight  and  incidental  a  thing  that  it  is  commonly  overlooked  as 
being  a  mechanical  process  at  all.  In  the  case  of  the  manufacture 
of  the  railroad-train  it  is  so  complex  and  important  that  a  whole 
string  of  factories  is  required  for  its  performance. 

The  same  thing  is  true  of  the  different  sorts  of  energetic 
cycles.  The  processes  DA  and  BC  are  variations  in  the  ex- 
tensity  of  energy:  The  detachment  of  the  working  water  from 
unity  with  the  earth,  in  the  first  place,  and  its  reconsolidation 
with  the  earth  in  the  second.  The  processes  AB  and  CD  are 
variations  in  the  intensity  of  the  mass-pair,  after  its  formation 
or  destruction  by  the  variations  in  extensity.  In  the  hydraulic 
cycles  these  variations  in  extensity  are  commonly  overlooked,  as 
essential  parts  of  the  cycle ;  and  yet  they  are  difficult  enough  to 
perform  properly.  The  water  must  be  gotten  into  the  overshot 
wheel  with  as  little  free  fall,  impact  and  friction  as  possible,  and 
gotten  out  again  in  the  same  way.  In  the  turbine  or  Pelton 
wheel  the  supply  of  motion-bearing  water  must  be  transferred  to 
the  vanes  with  as  little  impact  and  friction  as  possible,  and  the 
water  must  leave  the  vanes  for  the  tail-race  in  the  same  way. 
The  skill  with  which  these  extensity-varying  processes  are  car- 
ried out  often  has  as  much  to  do  with  the  efficiency  of  the 
wheel  as  has  the  skill  with  which  the  intensity-varying  processes, 
by  which  the  height  or  velocity  of  the  water  is  reduced  within  the 
wheel,  are  performed — and  in  the  former  case  usually  much  more. 

It  is  in  this  careful  way  that  the  most  familiar  mechanical 
processes  of  the  shop  and  the  power-house  must  be  analysed  and 
understood,  if  they  are  to  form  a  means  for  understanding  the 
more  obscure  cycles  of  molecular  mechanics.  For,  as  the  different 
forms  of  energy  and  methods  of  cyclical  action  are  taken  up,  one 
after  another,  hardly  any  two  will  be  similar  in  their  external 
appearance.  One  will  emphasize  the  DA-process,  or  the  prepara- 
tion of  the  mass-pair ;  as  in  the  building  of  the  railway-train  and 
road.  Another  will  emphasize  the  AB-process  of  dropping  the 
intensity  under  control,  as  in  the  turbine  water-wheel.  Another, 
like  the  overshot  water-wheel,  will  be  excluded  from  practica- 
bility by  its  exaggeration  of  the  CD,  or  empty-bucket,  work,  in 


THE  ENERGETIC  CYCLE  165 

proportion  to  its  net  work  DABC ;  while  in  the  Pelton  wheel  the 
CD-work  will  be  entirely  absent.  Yet  the  underlying  principles 
are  in  every  case  the  same.  And  as  the  study  of  thermodynamic 
cycles  is  reached  this  dissimilarity  of  appearance  and  identity  of 
principle  becomes  most  marked. 

The  first  step  toward  these  more  obscure  cycles  is  to  note  that 
the  mathematical  statements  concerning  the  static  and  kinetic 
hydraulic  cycles  already  given  are  not  quite  correct.  That  for 
the  overshot  water-wheel  assumed  that  the  weight  of  the  water 
would  be  constant  for  all  heights  above  the  earth,  which  is  not 
quite  true.  That  for  the  Pelton  wheel  assumed  that  the  kinetic 
energy  was  proportional  to  the  square  of  the  velocity,  whereas 
the  sum  of  the  masses  present  must  be  inserted  as  a  divisor.  The 
true  expressions  for  the  intensity  itself  appeared  in  Equations  24 
and  25. 

In  order,  then,  to  have  Fig.  10  become  exactly  and  generally 
true,  its  axes  must  be  understood  as  measuring,  not  weight  and 
height,  or  mass  and  velocity-squared,  respectively,  but  the  true 
factors  of  intensity  and  extensity  of  energy,  respectively.  The 
ordinates  must  measure  intensity,  the  abscissae  extensity,  of 
whatever  form  of  energy  is  considered.  That  the  mind  may 
effect  this  transfer  of  ideas  from  the  similes  already  employed, 
four  mental  substitutions  from  the .  approximate  to  the  exact 
must  be  made.  These  are: 

1.  The  substitution  of  the  exact  extensity-f actor,  or  degree 
of  mass-pairing  or  mass-product,  for  its  approximate  manifesta- 
tion, weight. 

2.  The  substitution  of  the  exact  spacial  intensity-factor,  or 

propinquity  =  c  — ,  for  its  approximate  manifestation,  height. 
S 

3.  The  substitution  of  the  exact  kinetic  intensity- factor,  or 
vclocity-squared-divided-by-mass,  for  its  approximation,  velocity- 
squared. 

4.  For  heat-energy,   the  concept  of  Fig.   10  as   portraying 
both    of    these   intensity-factors    simultaneously,    for   any   given 
complex  mass-system,  instead  of  merely  one  alone. 

This  careful  line  of  attack  upon  the  mysteries  of  the  thermo- 
dynamic cycles  has  been  long  and  perhaps  tedious ;  but  the  fruit 
is  worth  the  trouble.  The  typical,  or  pure,  thermodynamic  cycle, 
the  Carnot  cycle,  now  appears  as  nothing  more  mysterious  than 


166  ENERGY 

the  cycles  of  the  old-fashioned  overshot  and  the  newer  Pelton 
water-wheel,  in  intricate  combination;  somewhat  disguised  by 
the  number,  minuteness  and  intricacy  of  the  motion-bearing  mass- 
pairs  which  supply  the  energy,  it  is  true,  but  in  no  wise  more 
mysterious  in  its  principle  of  operation. 

In  1824  Sadi  Carnot,  in  his  "Reflections  upon  the  Motive 
Power  of  Heat,"  laid  down  these  rules  for  the  performance  of  a 
thermodynamic  cycle  with  the  maximum  possible  efficiency  be- 
tween any  given  temperature-limits  :* 

1.  The  heat  must  be   taken   in  isothermally,   at   the   tem- 
perature of  supply ; 

2.  The  heat  must  then   drop  its   temperature,   performing 
work,  isentropically ; 

3.  The  heat  then  rejected  must  pass  to  the  refrigerator,  or 
absorbent,  isothermally,  at  the  temperature  of  the  refrigerator; 

4.  The  remnant  of  heat  still  left  must  be  raised  to  the  tem- 
perature of  the  original  supply  isentropically,  absorbing  work. 

The  four  processes  thus  defined  are  seen  to  constitute  two 
pairs.  Each  pair  consists  of  examples  of  one  of  the  two  thermo- 
dynamic processes  which  were  defined,  in  Chapter  XI,  as  the 
only  basic  ones.  One  of  these  was  the  isothermal  variation  of 
entropy.  The  other  was  the  isentropic  variation  of  temperature. 
And  these  two  fundamental  thermal  processes  are  identical  with 
the  two  fundamental  mechanical  processes  which  were  defined 
and  numbered  on  page  163. 

That  is  to  say,  the  four  processes  of  the  Carnot  cycle  may  be 
represented  by  Fig.  10  with  perfect  accuracy,  if  the  two  axes  be 
considered  as  measuring  absolute  temperature  in  the  vertical  case 
and  entropy  in  the  horizontal.  Fig.  10  represented  the  kinetic 
mechanical  cycle  with  equal  accuracy  when  these  two  axes  were 
considered  as  measuring  intensity  of  space  and  motion  in  the 
vertical  case,  and  extensity  of  subdivision  of  matter  in  the  hori- 
zontal case.  The  only  things  needful  to  be  kept  in  mind,  in  order 
to  identify  the  two  cycles  completely,  are  three  in  number.  In 
the  first  place,  the  distinction  between  radial  and  tangential 
mechanical  action  within  the  molecule  must  be  understood. 

*The  language  has  been  modernized  into  accord  with  the  terms  already 
made  familiar  to  the  reader  in  the  preceding  pages.  Carnot's  language  was 
no  less  clear  and  explicit,  except  that  he  confused  the  ideas  of  2  M  and 
2  MM — a  confusion  which  is  still  widespread  among  teachers  of  energetics 
today. 


THE  ENERGETIC  CYCLE  167 

Work  can  be  performed  only  by  radial  mechanical  or  molecular 
energy. 

In  the  second  place,  comes  the  simple,  but  confusing,  question 
of  locality  in  which  the  several  processes  are  performed. 

In  the  third  place,  any  thermodynamic  cycle,  such  as  the 
Camot,  comprises  both  forms  of  mechanical  cycle:  the  kinetic 
and  the  static,  in  its  molecular  action — just  as  a  cannon-ball  at 
the  top  of  its  trajectory  will  do  work  in  virtue  of  both  its  height 
above  the  earth  and  its  horizontal  velocity  too. 

As  to  locality,  in  the  hydraulic  cycle,  for  instance,  the  cycle 
naturally  starts  with  the  process  DA:  the  acquisition  of  energy- 
bearing  mass.  No  thought  is  taken  as  to  how  nature  may  have 
prepared  this  natural  supply  of  energy-bearing  mass.  Those 
processes  lie  far  outside  our  water-wheels,  and  we  overlook 
them.  In  the  thermodynamic  cycles,  on  the  other  hand,  the 
preparation  of  the  energy-bearing  mass  occurs  within  our  power- 
houses, although  not  within  the  engines.  The  preparation  of  the 
steam  in  the  boiler-room  must  be  cared  for  and  understood,  as 
well  as  its  work-performance  in  the  engine-room.  In  the  boiler- 
room  the  heat  conducted  through  the  boiler-shell  acts  tangentially 
upon  the  water-molecule's  particles.  At  first,  as  the  water  is 
heating,  the  breaking  up  of  the  molecule  by  this  action  is  quickly 
annulled  by  its  compression  into  liquidity,  by  the  superior  pres- 
sure acting  on  the  surface  of  the  water.  As  each  molecule 
attains  sufficient  internal  whirling  velocity,  however,  so  that  its 
centrifugal  forces  are  able  to  overcome  these  external  forces, 
the  water-molecule  pops  into  a  steam-molecule.  In  order  to  give 
it  its  separateness,  while  at  the  same  time  conserving  its  same 
centrifugal  force  of  whirling,  or  its  "pressure,"  as  we  should  say, 
with  enormous  increase  in  volume — a  task  similar  to  elevating  a 
cannon-ball  to  the  highest  point  of  its  trajectory  while  conserving 
its  original  muzzle-velocity — a  large  quantity  of  energy  is 
absorbed,  in  latent  form.  The  steam-molecule,  thus  charged  with 
kinetic  energy  in  the  form  of  sensible  heat,  or  temperature,  and 
with  space-energy  in  the  form  of  volume  and  latent  heat  of 
vaporization,  is  then  carried  over  to  the  engine,  where  it  gives 
up  both  forms  of  energy,  in  part,  to  the  piston. 

The  piston,  it  is  true,  is  adapted  only  for  the  absorption  of 
the  kinetic  energy,  or  temperature-heat ;  but  while  the  kinetic 
intensity  of  the  steam-heat  is  no  greater  than  that  of  water  of 


168  ENERGY 

equal  temperature,  the  extent  of  kinetic  energy,  or  the  number 
and  mass  of  the  projectiles  which  are  in  a  condition  to  impinge 
upon  the  piston  has  been  vastly  increased  by  the  vaporization. 
A  water-molecule  engaged  in  work-performance  is  like  a  battle- 
ship firing  one  gun  at  a  time.  The  steam-molecule  is  like  the 
same  ship  firing  a  broadside.  The  intensity  of  projectile-energy 
(not  the  "intensity  of  fire,"  as  artillerists  use  the  term)  is  the 
same  for  one  gun  as  for  twenty  of  the  same  calibre.  For  the 
broadside  the  muzzle-velocity  is  no  greater,  but  the  effectiveness 
of  fire— the  "heft"  of  the  blow  against  the  enemy — is  far  greater. 
The  oft-heard  but  mistaken  argument  that  steam-heat  is 
inefficient  for  work-performance  because  of  the  latent  heat  in 
the  exhaust  is  exactly  parallel  to  the  possible  objection  to  broad- 
sides as  a  waste  of  ammunition.  If  the  enemy  be  present  in 
form  fit  to  absorb  a  broadside,  it  is  vastly  more  efficient  to 
employ  broadsides  than  a  dribble  of  single  fire.  The  work  is 
done  promptly,  powerfully  and  efficiently.  And  in  building  our 
steam-engines  there  is  no  difficulty  in  making  the  pistons  fit  to 
stand  the  broadsides.  The  powerful  steady  pressure  of  the  latent 
heat  of  the  steam  is  a  vastly  more  useful  laborer,  taking  the  run 
of  all  conditions,  than  is  the  highly  intense,  but  evanescent,  pres- 
sure of  the  gas-engine  cycle.  The  mere  fact  that  latent  heat 
costs  something,  as  do  broadsides,  has  little  to  do  with  the  case. 
Many  of  our  large  gas-engine  builders  are  to-day  modifying  their 
gas-producer  accessories  in  their  power-houses  so  that  their 
engines  may  have  more  entropy  to  work  with ;  though  to  tell 
them  that  this  was  what  they  are  unconsciously  doing,  or  that 
entropy  were  worthy  of  any  engineer's  serious  study,  would 
probably  fill  them  with  astonishment  and  disdain.* 

*Not  only  is  it  not  true  that  latent  heat  is  inimical  to  efficiency  of  work- 
performance,  but  it  is  true  that  no  other  form  of  heat  is  equally  efficient. 
Thus,  using  water  and  steam  as  the  illustrative  working-substance,  Fig.  B 
shows  the  Rankine  cycle  of  the  ordinary  steam-engine,  using  saturated 
steam,  including  the  heating  of  the  feed-water  in  the  boiler.  The 
heating  of  the  feed-water  is  shown  by  the  isomorphic  curve  AB,  the 
vaporization  of  the  hot  water  by  the  isothermal  BC,  the  work-performance 
in  the  engine-cylinder  by  the  isentropic  CD,  and  the  condensation  in 
the  condenser  by  the  isothermal  DA.  The  total  area  ABCDA  measures 
the  heat  converted  into  work.  Of  this,  that  derived  from  the  sensible 
heat  of  the  water  is  the  triangular  area  ABE;  that  derived  from 
the  latent  heat  of  the  steam  is  the  rectangle  BCDE.  From  this  it  is 
obvious  that,  for  equal  ranges  of  temperature  and  entropy,  the  efficiency 
of  sensible  or  isomorphic  heat  for  work-performance  is  less  than  half  that 
of  latent  heat. 


THE  ENERGETIC  CYCLE 


169 


As  the  molecule  expands  against  the  piston,  doing  work,  it 
loses  kinetic  energy  under  constancy  of  mass-pairing;  that  is  to 
say,  the  fastest  moving  particles  are  slowed  down,  by  rebound 

Indeed,  could  we  build  practicably  an  engine  which  would  follow  the 
cycle  BCDE,  using  only  latent  heat,  we  should  have  it  operating  upon  the 
Carnot  cycle,  the  one  cycle  of  maximum  efficiency. 


FIG.    B. 

Again,  compare  with  the  above  the  Beau  de  Rochas  cycle  of  the  ordi- 
nary gas-engine,  as  shown  in  Fig.  C.  Herein  the  total  temperature-range, 
from  A  to  C,  is  usually  some  3600°  F.  Since  the  absolute  temperature  at 
A  is  about  600°,  the  best  efficiency  obtainable  from  the  temperature-range 
available  would  be  3600-^-4200=0.86.  The  efficiency  of  the  Rankine  cycle, 


FIG.    C. 

under  common  ranges  of  pressure  and  temperature  in  condensing  engines, 
or  from  about  600°  to  800°  absolute,  would  be  similarly  computed  as  nearly 
0.25.  Of  this  25%  available  the  steam-engine  actually  develops  some 
three-fifths,  or  15%.  But  of  the  86%  available  for  the  gas-engine  the  latter 
seldom  exceeds  about  three-tenths,  or  26%.  The  difference  is  due  entirely 
to  the  poor  form  of  the  gas-engine  cycle.  The  trouble  is  that  its  heat  is 
all  isomorphic  or  sensible  in  form.  In  other  words,  were  it  practicable 
to  build  a  heat-engine  having  the  good  points  of  the  gas-engine,  but  using 
latent  heat  instead  of  isomorphic  heat,  its  efficiency  would  be  about  twice 
that  of  the  best  standard  gas-engines. 


170  ENERGY 

from  a  retreating  piston,  but  without  any  direct  tendency  to 
diminish  their  number.  That  could  be  done,  as  was  shown  from 
the  elementary  mass-pair,  only  by  a  tangential  retardation  at 
apastron  (the  point  of  "contact"  with  the  piston).  The  loss  of 
radial  component  against  the  piston — which  component  is  the 
only  one  which  motion  of  the  piston  might  affect,  and  which 
alone  constitutes  temperature — tends  only  to  decrease  the  eccen- 
tricity of  orbit  and  increase  the  periastron  distance.  The  satellitic 
particles  change  their  orbits  from  cometary  forms,  which  whirl 
closely  about  the  central  nucleus  and  then  shoot  into  space  with 
great  and  almost  radial  velocity,  to  orbits  like  those  of  the  outer 
planets :  slow,  remote  and  almost  circular.  The  change  is  like 
that  from  a  child's  return-ball,  shooting  out  and  back  on  an 
elastic  thread,  to  the  same  ball  whirled  about  the  hand  on  a 
string  of  fairly  fixed  length.  And  the  "hot  body,"  as  the  swarm 
of  such  molecules  is  called,  becomes  expanded,  cooled,  rarefied 
and  of  decreased  expansive  (or  radial)  pressure. 

It  happens,  in  the  case  of  steam,  that  as  this  process  gpes  on 
the  form  of  molecular  structure  leads  to  a  lack  of  internal 
equilibrium,  so  that,  in  order  that  the  majority  of  the  molecules 
may  maintain  the  form  described,  a  few  are  forced  to  recondense 
into  water — partial  condensation  being  a  well-known  accompani- 
ment of  the  adiabatic  expansion  of  steam.  But  this  fact  is 
merely  an  accidental  characteristic  of  water.  With  ether,  for 
instance,  the  exact  opposite  is  true.  Adiabatic  expansion  leads 
to  superheat.  Such  facts  as  these,  therefore,  have  no  bearing 
upon  the  general  idea  that  work-performance  by  heat  is  nothing 
more  than  a  transfer  of  purely  mechanical  energy  from  the 
molecules  to  the  piston. 

For  instance,  in  the  nozzle  of  a  steam-turbine  of  the  de  Laval 
type — and  the  nozzle  is  the  only  portion  of  this  steam-turbine 
where  thermodynamic  action  takes  place — this  mechanical 
explanation  of  adiabatic  expansion  is  even  more  obvious.  Here, 
instead  of  a  retreating  piston  being  opposed  to  the  radially  flying 
particles,  nothing  is  opposed.  The  nozzle  is  merely  a  device  for 
removing  from  the  steam  all  resistant  pressure  in  one  particular 
direction,  the  forward  one.  The  flying  particles  which  happen  to 
depart  from  their  molecular  nuclei  in  other  directions  than  this 
meet  the  solid  walls  of  the  nozzle,  and  return  with  orbits  unal- 
tered. But  those  which  happen  to  fly  in  the  direction  of  the  axis 


THE  ENERGETIC  CYCLE  171 

of  the  nozzle  find  nothing  to  return  them ;  and  as  they  all  possess 
hyperbolic  orbits,  they  depart  from  home  forever.  Instead  of 
giving  up  their  radial  velocity  to  a  piston,  they  keep  it.  Only 
now  it  has  become  a  linear  velocity,  not  only  in  reference  to  the 
nucleus  but  also  in  reference  to  the  walls  of  the  nozzle.  The 
swarm  of  molecules  strings  out  into  a  long  procession,  like  a 
body  of  troops  defiling  into  column-formation;  and  although  the 
cross-section  has  diminished,  the  volume  has  increased.  The 
molecules  find  more  "elbow-room."  And  while  their  original 
radial  velocity  remains  unchanged,  relatively  to  each  other  it  has 
disappeared.  Only  the  tangential  component  remains,  giving  to 
the  orbits  the  same  wide  circularity  which  they  had  after  colli- 
sion with  the  retreating  piston. 

The  impinging  of  this  jet  of  molecules  upon  the  curved  vanes 
of  the  turbine-wheel,  and  the  performance  of  work  there  by  the 
loss  of  linear  velocity,  is  a  purely  mechanical  action,  alike  in 
principle  in  hydraulic  and  steam  turbines.  The  expanded  mole- 
cules pouring  forth  from  the  nozzle  against  the  vane  might  just 
as  well  be  so  many  microscopic  but  inflated  foot-balls — originally 
fed  into  the  boiler  as  so  many  solid  wads  of  leather,  and  there 
inflated  with  compressed  air  by  the  energy  of  the  furnace,  and 
which  have  now  escaped  with  a  vigor  of  motion  derived  from 
their  own  internally  stored  energy — as  expanded  steam-molecules, 
so  far  as  any  thermodynamic  action  upon  the  vane  is  concerned. 
It  is  true  that  in  actual  practice  thermodynamic  action  nearly 
always  trespasses  beyond  the  nozzle  and  modifies  the  vane-action 
in  ways  which  cannot  occur  in  hydraulic  turbines.  But  this  is  a 
thing  which  the  steam-turbine  designer  usually  seeks  to  avoid,  or 
to  reduce  to  a  minimum. 

Cycle-efficiency.  In  any  rectangular  cycle,  such  as  Fig.  10, 
the  energy  taken  in  is  proportional  to  the  initial  intensity,  I1? 
whatever  the  width  of  the  cycle  horizontally.  Usually,  in  nature, 
cycles  are  truly  rectangular  only  when  considered  of  infinitesimal 
width,  dX.  But  since  any  cyclical  area,  however  irregular  in 
outline,  can  be  divided  into  an  infinite  number  of  vertical  strips 
of  width  dX  between  parallel  sides,  there  is  always  a  definite 
initial  and  final  intensity  of  rectangular  cyclical  action,  for  each 
increment  or  decrement  of  energy  handled. 

The  energy  rejected  by  the  cvcle  is  similarly  proportional  to 
the  intensity  of  rejection,  I2.  Therefore  the  efficiency  of  a  rec- 


172  ENERGY 

tangular,  or  perfect,  cycle,  or  the  proportion  of  the  energy  taken 
in  which   undergoes   transformation,  is  given  by  the  expression 

v=i  (34) 


which  is  the  fundamental  equation  for  all  pure  energy-trans- 
formation. 

In  mechanical  cycles  these  initial  and  final  intensities  are  pro- 
pinquities, or  velocity-squared-divided-by-mass,  as  already  defined. 
In  thermodynamic  cycles  they  are  temperatures.  In  electro- 
mechanical cycles  they  are  voltages.  But  whatever  the  form  of 
energy  may  be  the  law  stated  in  Equation  34  holds  true. 

Since  the  initial  intensity,  in  natural  action,  can  never  be 
infinite,  nor  the  final  intensity  zero,  no  cyclical  action  can  ever 
convert  all  of  any  primary  fund  of  energy  into  a  secondary  form. 
It  will  be  shown  later  that  energy-transformations  are  known 
between  almost  every  two  of  the  many  known  forms  of  energy  ; 
but  in  none  of  these  many  or  diverse  cases  can  the  transformation 
be  complete.  Although  it  is  commonly  said  that,  whereas  heat 
cannot  be  changed  completely  into  work,  yet  work  can  be  changed 
completely  into  heat,  nevertheless  this  statement  is  not  true,  as 
will  be  shown  later  —  except  in  a  special,  local  and  approximate 
sense  which,  while  useful  enough  for  certain  purposes,  has  no 
place  in  generalities  concerning  energetic  action.  Indeed,  we  can 
acquire  no  proper  understanding  of  energetic  action  until  we 
understand  that,  not  only  is  all  of  any  energy-form  never  trans- 
formed into  another  form,  but  the  difficulty  of  further  continuing 
the  transformation  approaches  infinity  as  the  energy  still  awaiting 
transformation  approaches  zero. 

Reversibility.  Upon  the  surface  of  the  earth,  when  dealing 
with  solid  or  viscous  bodies,  only  the  purely  vertical  of  these 
energetic  processes  are  obviously  "reversible."  Therefore  the 
rectangular  cycle,  of  maximum  efficiency,  has  usually  been  defined 
in  terms  of  "reversibility."  But  in  truth,  speaking  more  gen- 
erally of  the  principles  of  energetic  action,  not  only  do  the 
vertical  processes  never  occur  in  purity,  or  more  than  temporarily 
reversed,  but  the  horizontal  ones  are  equally  reversible  —  as  will 
be  brought  out  in  Chapter  XV.  As  a  temporary  aid  to  the 
student  energetic  reversibility  is  a  useful  idea;  but  as  a  cosmic 
principle  it  must  be  handled  with  great  care. 


THE  ENERGETIC  CYCLE  173 

Such,  in  general,  is  the  pure  energetic  cycle.     It  consists  of 
four  processes,  viz : 

I.  The  subdivision  of  mass,  at  a  high  degree  of  intensity. 
This  subdivision  may  occur  bodily,  as  in  splitting  off  from  the 
mill-pond    the    energy-bearing    water    which    enters    the    water- 
wheel;  or  it  may  occur  indirectly,  as  when  a  molecule  of  water 
within  a  steam-boiler  is  subdivided  by  tangential  contact  with  the 
rapidly  moving  molecules  of  the  hotter  boiler-shell.     In  either 
case    the   work-performer    (the    water-wheel   or    steam-engine) 
receives  mass  which  may  be  called  "energy-bearing"  because  it 
is  in  a  state  of  subdivision,  with  an  intensity  of  either  separation 
or  of  motion  (or  both,  as  in  the  case  of  steam)  between  every 
two  particles  formed  by  this  subdivision. 

II.  The  loss  of  intensity  under  constancy   of  subdivision. 
In  water-wheel  or  engine-cylinder  or  steam-turbine  nozzle  alike, 
the  number  and  size  of  mass-pairs  at  work  is  not  affected  by  the 
performance  of  work.    It  is  merely  their  intensity  of  relationship 
which  is  affected. 

III.  The  consolidation  of  mass  at  a  lower  degree  of  intensity, 
the  lowest  which  it  is  practicable  to  attain  by  work-performance. 
In  the  water-wheel  this  consists  in  dumping  the  now   useless 
water  into  the  tail-race.     In  the  steam-engine  it  is  the  condensa- 
tion of  the  steam  in  the  condenser. 

IV.  The  raising  of  intensity  of  the  now  partially  consolidated 
mass,  under  constancy  of  subdivision,  to  the  original  intensity — 
which,  because  of  the  consolidation,  takes  less  energy  than  was 
given  out  in  the  third  process.    This  is  the  most  obscure  of  all  of 
the  processes,  because  man  usually  leaves  nature  to  do  it  for  him, 
and  she  works  in  obscure  and  intricate  ways. 

The  performance  of  these  four  essential  processes  requires 
the  presence  of  four  essential  pieces  of  apparatus,  viz : 

I.  A  supply  of  energy-bearing  mass  of  high  intensity  of 
energy:  the  mill-pond  for  the  water-wheel,  the  steam-boiler  and 
furnace  for  the  steam-engine,  the  chemically  stored  energy  in  the 
explosive  gases  of  a  gas-engine  charge.  In  the  first  case  the 
mass  is  transferred  bodily  from  source  to  machine ;  in  the  second 
case  it  is  merely  the  subdivision  which  is  transferred,  by  "con- 
tact." In  the  third  case  occurs  no  transfer  at  all,  of  either  matter 
cr  entropy.  The  chemical  subdivision  between  carbon,  hydrogen 


174  ENERGY 

and  oxygen  is  transformed   directly   into   thermal   subdivision, 
within  the  molecule,  upon  ignition. 

II.  A  device  for  dropping  this  intensity  under  control:    in 
an  overshot   wheel   the   vertical   motion  of   the  buckets;   in   a 
turbine  the  curvature  and  motion  of  the  vanes;  in   a  steam- 
engine  the  retreat  of  the  piston  under  pressure;  in  the  steam- 
turbine  nozzle  the  open  end  and  conical  walls,  with  the  curved 
vanes  beyond. 

III.  A  device  for  absorbing  the  waste  energy  of  lowered 
intensity:    in  the  water-wheel  the  tail-race;  in  the  steam-engine 
the  condenser,  with  its  stream  of  cold  water. 

IV.  A  means  for  the  return  of  the  rejected  and  consolidated 
mass  to  its  original  condition:   in  the  water-wheel  the  sunheat, 
clouds  and  rain ;  in  the  steam-engine  the  feed-pump  and  furnace- 
heat.     It  is  this  process  which  man,  in  all  his   prime-movers, 
leaves  to  nature  to  perform  for  him.     Even  in  the  case  of  the 
steam-boiler  it  is  nature  who  is  relied  upon  to  furnish  the  coal 
to  keep  at  hot. 


CHAPTER  XIV. 

REVERSED  AND  IRREGULAR  CYCLES. 

The  cycle  which  was  described  in  the  preceding  paper, 
whether  relying  upon  a  water-wheel  or  a  steam-engine  for 
illustration,  was  considered  always  as  passing  through  the  four 
processes  in  an  order  which  followed  a  clockwise  direction  of 
passage  around  the  diagram.  Such  an  order  of  procedure  leads 
always  to  the  drawing  upon  nature  for  a  supply  of  energy  of 
high  intensity  of  one  form,  to  the  rejection  of  a  remnant  of 
that  energy  (with  all  the  mass,  or  mass-pairing,  which  carried 
it)  at  a  lower  intensity,  and  the  supply  to  the  operator  of  work 
at  a  usefully  high  intensity. 

It  is  obvious,  therefore,  that  cyclical  action  is  devoted  to  the 
transformation  of  energy,  from  one  form  which  nature  supplies 
to  another  which  man  desires.  The  first  of  these  forms  will  be 
called  the  primary  form  and  the  other  the  secondary  form.  In 
the  case  of  the  overshot  water-wheel  the  primary  form  is  static 
gravitational  energy,  or  space-energy,  between  mill-pond  water 
and  earth,  while  the  secondary  form  is  the  transient  energy  of 
the  wheel-shaft,  which  may  be  converted  into  any  of  many 
forms  within  the  mill.  In  the  hydraulic  turbine-nozzle  the 
primary  form  is  static  space-energy,  as  before,  but  the  secondary 
form  is  kinetic  water-earth  energy  of  jet.  In  the  vanes  them- 
selves of  the  hydraulic  turbine  the  primary  form  is  the  kinetic 
water-earth  energy  received  from  the  jet,  and  the  secondary 
form  is  the  transient  energy  of  the  wheel-shaft,  as  before.  In 
the  nozzle  of  the  steam-turbine  the  primary  form  of  energy  is 
steam-heat  of  high  temperature,  involving  both  space  and  motion- 
energy  between  the  particles  of  the  molecules,  as  already 
described,  while  the  secondary  form  is  kinetic  steam-earth  energy 
of  jet.  In  the  steam-turbine  vanes  the  primary  form  is  kinetic 
mechanical  energy  of  flying  steam,  while  the  secondary  form  is 
the  transient  energy  of  the  turbine-shaft. 

A  moment's  consideration  will  make  it  plain  that  there  is  a 
very  close  relationship  between  the  individual  links  of  any  of 

175 


176  ENERGY 

these  chains  of  cycles.  They  are  indissolubly  linked.  Thus, 
the  water-wheel  cycle  cannot  be  performed  unless  there  be  a 
device  for  absorbing  its  power.  Ordinarily  this  device  was  a 
pair  of  mill-stones,  which  absorbed  it  in  friction.  As  this  is  too 
indefinite  for  our  purpose,  let  it  be  supposed  that  the  water- 
wheel  drives  a  pump  which  draws  water  from  the  tail-race  and 
discharges  it  into  the  mill-pond.  Then,  if  the  picture  is  to  be 
complete,  no  water-wheel  cycle  could  be  drawn  without  drawing 
also  a  pump-cycle.  And  the  pump-cycle,  if  friction  is  to  be 
neglected  for  the  moment,  would  be  of  equal  area,  or  power, 
with  the  driving  cycle. 

But  the  pump-cycle  must  be  portrayed  in  a  reversed,  or 
counterclockwise  direction.  It  receives  mass-pairing  at  a  low 
intensity  of  energy,  raises  its  intensity,  discharges  it  into  the 
pond  (or  consolidates  it  with  the  earth),  and  then  goes  back  for 
more.  Such  a  cycle  would  be  shown  by  Fig.  10,  if  the  diagram 
be  traversed  in  the  direction  CBAD.  The  reversed  cycle,  there- 
fore, is  one  in  which  the  primary  energy  is  received  at  low 
intensity  and  the  secondary  form  discharged  at  high  intensity. 

It  is  merely  a  corollary  of  the  law  of  the  conservation  of 
energy  to  state  that  for  every  clockwise  or  direct  cycle  which 
takes  place,  there  must  be  performed  a  counterclockwise  or 
reversed  cycle  of  equal  area,  to  absorb  its  energy.  No  water- 
wheel  can  operate  without  a  pump  or  equivalent  to  absorb  its 
power. 

In  portraying  the  action  of  any  energetic  machine,  therefore, 
it  is  merely  a  matter  of  choice  whether  we  portray  the  direct 
or  the  reversed  cycle  which  occurs  there.  In  the  Pelton-wheel 
penstock  and  nozzle,  for  instance,  there  occurs  a  direct  cycle  of 
static  hydraulic  energy,  which  enters  at  high  head  and  leaves  at 
low;  but  there  is  simultaneously  performed  a  reversed  cycle  of 
kinetic  hydraulic  energy,  for  the  water  enters  at  a  low  velocity 
and  leaves  at  much  higher  speed.  The  cycles  are  like  coiled 
springs.  The  unwinding  of  the  gravitational  cycle  winds  up  the 
accelerative  cycle;  and  the  unwinding  of  the  accelerative  cycle, 
within  the  wheel,  a  moment  after,  winds  up  some  equivalent 
cycle  of  a  further  form. 

Thus  nature  works  in  an  endless  chain  of  transformations, 
by  cyclical  action,  one  thing  giving  up  its  strength  that  the  next 
may  live,  and  the  next  called  upon  a  moment  later  to  undergo 


REVERSED  AND  IRREGULAR  CYCLES          177 

the  same  process  of  reproduction.  In  the  power-house  is  seen 
quite  a  chain  of  such  cyclical  actions.  The  energy  enters  in  the 
form  of  high-intensity  chemical  energy,  in  the  coal-pile  and 
atmosphere.  This  cycle  unwinds  in  combustion,  winding  up 
simultaneously  the  cycle  of  thermal  intensity,  in  high-temperature 
heat.  This,  in  reality,  unwinds  again  as  its  energy  is  radiated 
and  conducted  through  the  boiler-shell,  though  the  fact  cannot 
be  seen  unless  it  is  analysed  mechanically;  and  simultaneously 
it  winds  up  a  similar  cycle,  though  a  slower  and  more  massive 
one,  in  the  heat  of  the  steam.  This  cycle  unwinds  again  in  the 
engine-cylinder,  winding  up  the  mechanical  cycle  of  the  engine- 
shaft.  Or,  if  a  steam-turbine  be  used,  there  are  two  transforma- 
tions within  the  engine,  one  in  the  nozzle  or  guide-blades  and  the 
other  in  the  vanes  of  the  rotor. 

So  the  energy  might  be  followed  upon  its  way,  through  elec- 
trical and  other  forms ;  and  everywhere,  so  far  as  can  be  seen, 
the  chain  of  unwinding  and  winding  up  continues  endlessly. 

In  human  affairs  man  is  usually  more  interested  in  the 
unwinding,  or  direct,  cycles;  for  he  needs  power  more  than  he 
needs  absorbents  of  power.  But  to  a  wide  extent  he  also  uses 
the  latter.  Pumps,  elevators  and  refrigerating-machines  all 
belong  to  this  class.  The  cycle  of  the  pump  and  the  elevator  is 
as  simple,  though  reversed,  as  that  of  the  overshot  water-wheel, 
and  needs  no  explanation.  The  cycle  of  the  refrigerating- 
machine  is  much  more  obscure,  however.  For  its  details  the 
reader  must  study  the  refrigerating-machines  in  detail.  All  that 
can  be  said  here  is  that  such  a  machine  pumps  low-temperature 
entropy  (not  heat)  from  the  cold-storage  room,  or  the  water  to  be 
frozen,  and  discharges  it  at  high  temperature  into  any  convenient 
waste,  exactly  as  a  pump  picks  up  low-level  water  and  discharges 
it  at  a  higher  level.  The  analogy  is  scientific  and  exact.  It  has 
already  opened  to  our  understanding  wide  fields  of  most  useful 
progress  in  the  arts,  which  a\vait  only  a  slight  advance  in  our 
economic  organization  to  make  thoroughly  practicable.  We  refer 
to  the  supply  of  heat  for  buildings  upon  a  large  scale  by  literally 
pumping  up-temperature  the  low-temperature  entropy  which 
surrounds  us  during  the  winter  months — perfectly  good  entropy, 
all  of  it;  only  a  little  too  low  on  the  temperature-scale  for  our 
use.  Now  we  rely  upon  heat  to  heat  our  homes.  Soon  we  shall 
reply  upon  power  for  that  purpose;  not  by  converting  it  into 


178  ENERGY 

heat,  in  friction,  but  in  pumping  out-door  heat  far  enough  up- 
temperature  to  be  agreeable  to  human  nerves.* 

Irregular  Cycles.  So  soon  as  attempt  is  made  to  carry  out 
in  practice  any  of  the  cycles  described  in  connection  with  Fig.  10, 
it  appears  that  it  is  impossible  to  do  so,  in  purity.  Nature  never 
performs  either  of  the  two  processes  which  were  described  as 
the  basic  energetic  ones,  the  horizontal  and  vertical  ones  respect- 
ively, in  purity.  Each  is  always  adulterated  by  the  presence  of 
the  other  to  some  degree. 

To  explain,  let  Fig.  n  represent  an  energetic  field  in  which 
the  sources  and  absorbents  of  energy  are  situated  at  the  levels 
It  and  I2,  respectively.  These  may  be  imagined  as  mill-pond  and 
tail-race,  if  desired;  but  the  diagram  applies  equally  to  any  of 
the  more  obscure  forms  of  energy.  If,  in  such  a  field,  it  is 
desired  to  operate  a  direct  cycle,  such  as  DABC,  it  proves  to  be 
impossible,  with  exactness.  In  order  to  effect  the  transfer  from 
the  sources  of  supply  to  the  machine,  the  energy  must  be  taken 
in,  in  whole  or  in  part,  at  levels  somewhat  below  DA,  as  along 
da.  Similarly,  it  must  be  discharged  at  levels  somewhat  higher 
than  BC,  as  along  be.  The  area  of  the  cycle  dabc  is  therefore 
less  than  that  of  the  cycle  DABC,  the  cycle  described  by  Carnot 
as  the  one  of  maximum  efficiency. 

Similarly,  the  attempt  to  carry  out  a  reversed  cycle,  such  as 
CBAD,  between  these  intensity-levels  develops  the  fact  that  the 
energy  must  be  taken  in  at  some  level  below  CB,  as  along  CGB, 
and  discharged  at  some  higher  level  than  AD,  as  along  AHD. 
This  cycle  therefore  absorbs  more  power,  as  measured  by  its 
area  CGBAHD,  than  is  stored  by  it  usefully,  as  measured  by  the 
area  CBAD.  Its  efficiency,  too,  is  below  the  maximum  possible 
with  rectangular  cycles. 

The  irregular  direct  cycle  dabc  is  like  that  of  a  water- 
wheel  into  which  the  water  leaks  while  the  buckets  are  still 
rising  or  after  they  have  begun  their  fall,  or  out  of  which  the 
water  leaks  before  the  fall  is  completed  or  after  return  has 
begun.  The  irregular  reversed  cycle  CGBAHD  is  like  a  pump 

*Tell  a  man  who  is  spreading  his  hands  before  a  blazing  fire  on  a  win- 
ter's night  that  what  he  is  absorbing  and  enjoying  is  not  temperature,  but 
entropy,  and  the  speaker  would  probably  suffer.  Nevertheless,  it  is  true. 
Man  takes  almost  enoup-h  bodily  comfort,  as  well  as  industrial  profit,  out 
of  this  much  abused,  ignored  and  contemned  drudge,  entropy,  to  justify 
the  proverbial  saying  that  "ignorance  is  bliss." 


REVERSED  AND  IRREGULAR  CYCLES 


179 


which  draws  in  at  a  level  below  that  of  supply,  and  discharges 
above  the  waste-level.  It  is  like  that  of  a  hod-carrier  who  should 
be  set  to  carrying  bricks  from  the  street-level  to  the  third  floor, 
but  who  took  his  hod  into  the  cellar  to  drop  the  bricks  into  it, 
then  carried  it  to  the  fourth  floor,  and  then  dumped  the  bricks 
back  to  the  third  floor. 

From  these  considerations  has  been  enunciated  the  general 
principle  of  energetics  that  the  rectangular  cycle  of  pure  pro- 
cesses (as  defined  elsewhere  in  these  papers)  is  the  cycle  of 
maximum  efficiency.  Carnot  was  the  first  to  define  this  law, 
and  he  spoke  in  terms  of  thermodynamic  cycles  only ;  but  the 
material  for  its  application  to  mechanical  cycles  has  existed  ever 
since  the  work  of  Newton,  and  that  for  electrical  applications 
ever  since  that  of  Faraday  and  Ohm. 


FIG.   II. 

If  a  direct  cycle  working  between  the  limits  Ij  and  I2  should 
be  set  to  operate  a  reversed  cycle  (as  always  occurs  in  nature), 
it  is  plain  that  the  direct  cycle  must  take  some  irregular  form, 
such  as  dabc,  which  is  inscribed  within  the  rectangle  DABC. 
The  reversed  cycle  which  is  operated  from  it  must  in  turn  be 
inscribed  within  the  rectangle  DABC.  Therefore  the  limits  of 
intensity  between  which  the  reversed  cycle  is  finally  effective 
must  be  continually  lower  and  lower  ones,  such  as  I3  and  I4, 
which  lie  between  Ix  and  I2.  It  is  this  fact  which  has  led  to  the 


180  ENERGY 

broad  doctrine  that  the  availability  or  intensity  of  the  world's 
stock  of  energy  is  steadily  declining. 

Nevertheless,  the  statement  is  not  true.  It  was  based  upon 
too  obscure  a  form  of  cycle,  the  thermodynamic,  for  all  its 
bearings  to  be  clearly  seen.  So  soon  as  it  is  referred  to  a  form 
where  all  the  details  can  be  followed,  as  in  mechanical  energy, 
its  falseness  appears  as  unquestionable.  For  then  the  "degrada- 
tion of  availability"  turns  out  to  be  based  solely  upon  reference 
to  a  single  form  of  energy — upon  a  form  dictated  solely  by 
temporary  human  desire — and  not  upon  any  broad  natural 
principle.  Such  a  foundation  is  entirely  too  narrow  to  support 
a  fundamental  cosmic  law.  For  what  human  beings  preemi- 
nently desire  seems  to  be  rectilinear,  or  radial,  motion.  They 
have  little  use  for  tangential  motion,  except  as  it  can  be  con- 
verted into  rectilinear. 

Thus,  observe  a  steam-boat  ploughing  through  the  water. 
In  the  engines,  in  the  steam  within  them,  in  the  boat  itself, 
and  in  the  water  surrounding  the  boat,  is  an  intricate  mixture 
of  radial  and  tangential  motions.  Ultimately  it  all  becomes  vir- 
tually tangential  motion  within  the  water,  in  tiny  eddies  which 
constitute  hydraulic  resistance.  These,  later  on,  become  still 
tinier  molecular  eddies  called  low-temperature  heat.  But  one 
stage  in  the  progress  of  the  energy  to  this  destination  consists 
in  a  rectilinear,  forward  motion  of  the  ship.  Of  all  the  motions 
present,  this  alone  man  esteems.  He  uses  it  as  his  measure  of 
"efficiency."  But  nature  esteems  all  motions  equally,  for  she 
is  able  to  reconvert  the  tiny  eddies  into  rectilinear  motion  when 
she  wishes  to  do  so,  as  man  cannot. 

In  the  economy  of  nature  the  readjustment  of  equilibrium  is 
being  constantly  made  by  means  of  these  chains  of  direct  and 
reversed  cycles  of  energetic  action.  The  currents  of  energy  are 
fairly  continuous,  seldom  starting  or  stopping  abruptly.  The 
portions  of  matter  through  which  they  must  find  their  way, 
however,  are  limited  in  their  mass  and  dimensions.  Therefore 
each  portion  of  mass,  in  order  to  perform  its  allotted  task  of 
energy-transformation,  must  act  over  and  over  again.  This  it 
does  by  the  method  of  the  cycle. 

The  ubiquity  of  these  chains  of  cycles  is  too  great  for  com- 
prehension.  Wherever  energetic  action  occurs,  there  proceed 
these  chains  of  cycles.  We  have  twice  referred  to  that  visible 


REVERSED  AND  IRREGULAR  CYCLES          181 

in  the  series  of  machines  and  processes  incidental  to  the  modern 
power-house.  But  is  it  realized  that  each  tiny  molecule,  not  to 
mention  smaller  portions  of  matter,  in  all  this  apparatus  is  itself 
constantly  carrying  on  its  own  particular  chain  of  cycles,  peculiar 
to  itself?  It  is  the  integration  of  these  countless  hordes  of  less 
than  microscopic  cycles,  into  something  big  enough  for  man  to 
see,  which  constitutes  that  major  chain  of  phenomena  which 
characterizes  modern  power  development  and  distribution. 

But  the  power-house  and  its  accessories  are  but  a  tiny  instance 
of  nature's  broad  dependence  upon  cyclical  action.  For  a  single 
instance,  observe  the  interaction  between  vegetable  and  animal 
life.  All  vegetation  is  continuously  operating  a  thermochemical 
cycle  in  a  single  direction.  Drawing  high-intensity  radiant 
energy  from  the  sun  and  low-intensity  chemical  energy  from  soil 
and  air,  in  the  form  of  the  very  stable  chemical  compounds, 
water  and  carbon-dioxid,  respectively,  it  operates  a  direct  thermal 
cycle  to  keep  in  operation  a  reversed  chemical  cycle.  It  main- 
tains a  low  normal  temperature  automatically,  so  that  we  seek 
the  "cool  green  shade"  on  summer-days;  and  it  stores  up  chem- 
icals of  a  higher  intensity,  in  the  form  of  starch,  sugar  and 
similar  nutrients. 

In  apposition  with  these  cycles,  all  animal  life  maintains  their 
obverse.  Animals  absorb  the  starch  and  sugar  and  operate  a 
direct  chemical  cycle  in  their  degradation  into  carbon  dioxid  and 
moisture.  They  do  this  in  order  to  keep  in  operation  reversed 
mechanical  and  thermal  cycles,  resulting  in  animal  motion  and 
high-temperature  animal  heat. 

Thus  each  half  of  animate  existence  here  on  earth  both  sup- 
ports, and  at  the  same  time  properly  loads,  controls  and  balances, 
the  other.  Without  the  cooperation  of  the  other,  neither  could 
exist.  Either  starvation  or  apoplectic  surfeit  would  ensue.  « 


CHAPTER  XV. 

THERMAL  EQUILIBRIUM. 

In  summing  up  the  general  characteristics  of  mechanical 
energy,  in  Chapter  VI,  it  was  carefully  pointed  out  that  all 
energetic  conditions  of  matter  varied,  not  in  one  direction  only, 
from  an  absolute  zero,  but  in  both  directions,  from  a  central  or 
"mean  energetic"  condition.  It  was  pointed  out  in  detail  how, 
from  this  central  condition,  each  of  the  several  factors  of 
mechanical  energy — force,  velocity,  space  and  even  the  mass- 
factor  itself — vibrated  in  either  direction  indefinitely,  yet  limited 
elastically  in  stable  equilibrium.  For,  as  any  factor  proceeded 
away  from  the  mean  condition  it  engendered  forces  and  phe- 
nomena which  tended  always  to  resist  its  further  progress  and 
to  return  it  toward  centricity.  Thus,  excessive  velocity  resulted 
in  separation ;  but  separation  involved  the  storage  of  energy  in 
space-form  at  the  expense  of  velocity-form.  Excessive  separa- 
tion, on  the  other  hand,  annulled  the  velocity  which  permitted  it 
to  exist,  and  invited  the  regain  of  velocity  in  a  return  into  pro- 
pinquity. 

Thermal  Equilibrium.  If,  now,  attention  be  turned  to 
Fig.  12,  it  will  be  plain  that  thermal  energy  shows  every  indica- 
tion of  following  all  the  characteristics  of  mechanical  energy,  in 
its  range  from  the  unusually  cold  and  solid  conditions  of  matter, 
as  at  B,  to  the  unusually  hot  and  fluid  conditions,  as  at  H.  The 
only  difficulty  in  understanding  the  fact  lies  in  the  necessity  of 
comprehending  the  unusually  hot  extreme  in  terms  of  the 
mechanical  phenomena  of  the  earth's  surface;  which,  as  has  just 
been  pointed  out,  concern  themselves  with  matter  in  the  extreme 
opposite  condition,  unusually  hard  and  dense. 

In  the  first  place,  the  range  of  thermal  conditions  follows,  for 
every  substance,  some  such  a  curve  as  BCADEFGH.  This  curve 
exhibits  stability  of  equilibrium  at  every  point,  except  where  it 
is  interrupted  by  the  fields  of  instability  which  we  call  fusion, 
vaporization  and  chemical  dissociation  respectively.  Across  all 
such  gaps  thermal  conditions  must  jump  abruptly.  The  curve 

182 


THERMAL  EQUILIBRIUM  183 

is  asymptotic  to  the  axis  of  absolute  zero  of  temperature  XX 
at  the  left.  It  can  never  reach  it;  and  as  it  approaches  it  the 
increase  in  negative  entropy,  or  solidity,  becomes  increasingly 
great  with  each  step  nearer. 

The  other  limb  of  the  curve,  toward  H,  approaches  increas- 
ingly the  vertical  direction.  From  knowledge  yet  available,  which 
is  comprised  in  Equation  30,  it  cannot  be  said  explicitly  that  this 
limb  is  asymptotic  to  a  vertical  axis.  But  Equation  30,  it  must 
be  remembered,  is  based  upon  too  narrow  a  ground  to  be  extrapo- 
lated into  a  general  principle.  It  is  based,  first,  upon  an  assumed 
constancy  of  the  specific  heat;  yet  of  specific  heats  in  the  higher 
ranges  of  temperature  we  possess  very  meagre  knowledge.  It 
happens,  it  is  true,  that  our  most  recent  acquisitions  in  this 
direction  point  to  an  increasing  specific  heat  for  gases,  as  tem- 
peratures rise;  and  this  would  tend  to  maintain  the  obliquity  of 
the  curve  at  H.  But  the  specific  heat  of  water  also  rises  with 
the  temperature;  yet  this  fact  is  only  preliminary  to  a  stage,  the 
critical  temperature,  above  which  specific  heats  become  very 
much  smaller  and  the  isomorphic  curve  much  steeper. 

Finally,  reference  must  be  had  to  the  mathematical  concept 
of  the  "perfect"  gas,  toward  whose  attributes  gases  tend  as 
they  rise  in  temperature;  and  this  perfect  gas,  having  no  vis- 
cosity, could  be  represented  upon  Fig.  12  only  by  a  straight 
vertical  line.  For  all  of  these  reasons,  taken  in  connection  with 
the  fact  that  every  other  known  energetic  function  becomes 
asymptotic  at  its  either  end,  the  conclusion  cannot  be  escaped 
that  the  thermal  diagram  would  also  extend  its  upper  end  into  a 
real,  as  well  as  an  apparent,  asymptosy,  if  its  true  form  could 
be  known ;  although  it  is  impossible  now  to  define  either  its  true 
form  of  function  or  the  lateral  distance  of  its  axis  from  any  mean 
thermal  condition. 

Of  these  two  axes  to  which  thermal  conditions  are  asymp- 
totic, the  first  named  or  horizontal  one,  hitherto  known  as  that 
of  the  "absolute  zero  of  temperature,"  will  be  referred  to  here- 
inafter as  the  axis  of  absolute  solidity  of  matter,  where  exist 
no  fluidity,  no  elasticity,  no  expansivity  and  no  translucence.  To 
these  characteristics  might  be  added  no  temperature,  infinitely 
negative  entropy,  no  volume,  and  infinite  density.  But  whereas 
the  latter  list  is  meaningless  to  us,  we  have  an  idea  that  we  know 
what  the  former  qualities  signify ;  for  are  they  not  the  ordinary 


184  ENERGY 

attributes  of  solid  matter?  Yet  the  method  of  approach  to  the 
statement  just  given  was  chosen  in  order  to  make  it  clear  that 
this  axis  of  absolute  solidity  defines  mathematically  a  condition 
which  can  never  be  reached  in  nature.  Its  characteristics  are 
those  which  matter  never  can  exhibit. 

Of  the  two  axes  of  asymptosy,  the  other  or  vertical  axis  is 
that  of  the  so-called  "perfect,"  or,  as  it  will  be  called  herein- 
after, the  absolute,  gas.  In  this  condition  matter  possesses  per- 
fect fluidity  or  no  viscosity,  perfect  elasticity  or  no  disgregation- 
work,  and  perfect  translucence.  Incidentally  it  must  possess 
either  infinite  temperature  or  infinite  volume,  or  both,  or  its 
density  must  be  zero.  Viewed  mechanically,  its  orbits  must  be 
all  hyperbolic  and  none  elliptic;  its  mass  must  be  all  satellitic 
and  none  nuclear.  This,  too,  is  a  condition  which  may  be 
mathematically  defined — and  such  a  mathematically  defined  limit 
is  most  useful  to  the  understanding — but  it  is  a  condition  which, 
however  closely  approached,  may  never  be  reached  in  nature. 

Now  this  so-called  perfect  gas — as  if  anything  which  the 
Supreme  Intelligence  had  seen  fit  to  exclude  sweepingly  from 
the  universe  could  be  called  "perfect" — is  so  impossible  and  un- 
natural a  thing  that  the  writer  has  carefully  excluded  it  from 
all  of  his  teaching.  But  it  has  been  used  so  widely,  by  other 
teachers,  as  the  base  and  explanation  of  all  thermodynamic 
action,  that  it  must  be  mentioned  here  to  the  extent  of  putting 
it  in  its  proper  place. 

That  is  to  say,  there  exists  on  one  side  of  all  energetic  action 
(at  the  lower  left-hand  of  Fig.  12)  an  unattainable  limit  of 
deficit  of  internal  energy,  when  all  internal  motion  would  be 
purely  tangential,  or  circular,  and  the  pressure  zero ;  which  con- 
dition would  be  attainable,  if  at  all,  only  when  the  temperature 
were  absolutely  zero  and  the  quantity-factor  of  heat  infinitely 
negative;  in  which  condition,  if  ever  attainable,  the  substance 
would  constitute  a  "perfect" — or,  as  I  prefer  to  state  it,  an 
"absolute" — solid.  Such  a  state  of  affairs  would  be  exhibited 
by  the  point  B,  Fig.  12,  if  pushed  far  enough  to  the  left  (that 
is,  to  an  infinite  distance)  to  bring  it  into  coincidence  with  the 
axis  XX. 

On  the  other  side  of  all  energetic  action  (but  not  on  the 
opposite  side)  there  exists  an  unattainable  limit  of  surplus  of 
internal  energy,  when  all  internal  motion  would  be  purely  radial, 


THERMAL  EQUILIBRIUM  185 

or  rectilinear,  and  none  tangential ;  and  where  the  viscosity  would 
be  zero ;  which  condition  would  be  attainable,  if  at  all,  only  when 
the  temperature  were  infinite  and  the  quantity- factor  of  heat  at 
its  maximum,  as  shown  by  the  axis  NM.  Such  a  state  of  affairs 
would  be  exhibited  by  the  point  H  of  Fig.  12,  if  pushed  far 
enough  up  (that  is,  to  an  infinite  distance)  to  bring  it  into  coin- 
cidence with  the  axis  NiM  to  which  GH  is  asymptotic.  This 
axis  is  therefore  that  of  the  so-called  "perfect" — or,  as  I  prefer 
to  call  it,  the  "absolute" — gas. 

Between  these  two  unattainable  extremes,  or  purely  mathe- 
matical limits,  all  natural  action,  thermodynamic  or  otherwise, 
occurs — surging  back  and  forth  in  stable  equilibrium  about  some 
central  mean  energetic  or  thermal  condition.  In  so  far  as  it 
gains  approach  to  the  vertical  axis  of  zero  solidity,  or  gaseous 
perfection,  it  gains  fluidity,  elasticity,  expansivity  and  adaptability 
for  work-performance.  And  in  this  sense  it  is  true  that,  to  the 
extent  to  which  matter  possesses  those  qualities  which,  if  existing 
alone,  would  constitute  it  a  "perfect"  gas,  it  exhibits  faculty  for 
thermodynamic  labority.  But  equally  true  is  it  that  in  so  far  as 
it  gains  approach  to  the  horizontal  axis  of  zero  gaseousness,  or 
perfection  of  solidity,  it  gains  faculty  for  impact,  friction  and 
gravitational  work-performance ;  in  other  words — to  coin  a  par- 
allel term — for  dynamothermal  thermogy.  And  these  two  con- 
trasted faculties,  it  seems  to  me,  are  of  equal  importance  and 
deserve  equal  prominence  in  thermodynamic  discussion. 

Neither  limit  of  thermal  attributes  can  matter  ever  reach. 
Yet  each  of  these  limits  has  its  proper  use,  as  a  base  of  refer- 
ence. Some  writers  on  thermodynamics,  if  not  all,  have  preferred 
to  enter  the  topic  from  the  limit  properly  called  the  "absolute" 
gas  as  their  base  of  reference.  To  this  plan,  so  carried  out  that 
the  student  sees  the  true  relation  between  base  of  reference  and 
all  natural  action,  there  can  be  little  objection. 

The  writer  has  much  preferred,  however,  to  enter  the  topic 
from  the  other  limit,  that  of  the  "absolute"  solid,  by  studying 
first  the  elements  of  energetics  as  they  would  occur  between  two 
mass-portions  each  of  which  is  apparently  a  solid  unit.  This  is 
natural  because,  in  the  first  place,  the  growing  boy  has  dealt 
almost  entirely  with  solids,  in  work  and  play,  and  all  his  concepts 
of  mechanical  action  are  based  thereon.  In  the  second  place, 
the  mechanics  of  solids  is  already  reduced  to  an  exact  mathe- 


186  ENERGY 

matical  science,  which  is  supposed  to  be  taught  to  every  student 
of  engineering  before  he  meets  the  problems  of  thermodynamics. 
So  that  the  writer  regards  this  mode  of  entrance  to  thermo- 
dynamics as  far  preferable  to  that  via  the  "perfect"  or  "abso- 
lute" gas. 

The  one  plan  which  deserves  unlimited  condemnation,  how- 
ever, is  to  open  the  study  of  thermodynamics  with  the  concept 
of  the  absolute  gas,  using  actual  gases  as  "impure"  illustrations, 
and  to  leave  the  student  with  the  idea  that  what  likeness  to  the 
absolute  gas  can  be  found  in  nature  is  alone  thermodynamic 
action.  All  natural  action  is  thermodynamic,  to  whatever  extent 
heat  takes  part.  To  leave  the  student  with  his  concept  of  natural 
action  thus  equipped  with  a  roof,  in  the  form  of  a  knowledge  of 
the  gaseous  side  of  thermodynamics,  but  with  no  foundation  or 
adequate  frame-work  in  the  way  of  a  corresponding  knowledge 
of  the  solid  and  liquid  aspects  of  the  science,  is  what  is  being 
done  by  every  teacher  who  thus  misuses  the  mathematical  con- 
cept of  the  so-called  "perfect"  gas.* 

This  fundamental  fact  should  be  one  of  the  first  taught  the 
student  of  natural  science.  These  two  axes  define  the  boundaries 

The  following  language  concerning  this  situation  is  taken  from 
another  article  by  the  writer :  "We  know  of  no  gas  so  hot  or  so  rarefied 
and  so  'perfect'  that  it  loses  all  viscosity,  or  is  perfectly  elastic  and  free 
from  disgregation-work.  We  have  no  reason  to  think  that  any  such  a 
substance  could  ever  exist.  It  is  unobjectionable  to  refer,  upon  occasion, 
to  the  mathematical  hypotheses  of  the  'perfect  gas'  or  the  'perfect  solid* 
as  aids  in  argument,  as  has  just  been  done.  But  with  these  references 
should  always  go  the  explanation  that  these  two  hypotheses  constitute 
impossible,  supernatural  extremes,  between  which  vibrate  all  known 
natural  conditions  of  matter;  and  science  should  have  just  as  little  to 
do  with  the  supernatural  as  possible. 

"But  when  reliance  upon  the  perfect-gas  hypothesis  goes  so  far  as 
to  make  it  the  center  and  foundation — the  immediate  beginning,  rather 
than  the  remote  and  unattainable  horizon — of  all  thermodynamic  ^ study, 
as  is  now  very  widely  done,  the  writer  feels  impelled  to  rise,  in  the 
name  of  nature  and  common  sense,  to  denounce  the  practice.  His  experi- 
ence in  teaching  thermodynamics  has  found  this  method  so  universally 
confusing  to  the  engineering  student,  who  above  all  others  needs  acquaint- 
ance with  nature  rather  than  with  disembodied  and  supernatural  hy- 
potheses, and  so  belittling  to  the  dignity  of  the  instructor  posing  as  a 
Guide  to  the  Truth,  that  he  can  express  his  feelings  only  ^  by  a  free 
quotation  fr<wn  Mr.  Burgess's  'Purple  Cow' — with  apologies  to  the 
Cow,  as  well  as  to  Mr.  Burgess: 

'I  never  saw  a  Perfect  Gas. 
I  never  hope  to  see  one. 

But  I  can  tell  you,  as  we  pass, 
I'd  rather  see  than  be  one/  " 


THERMAL  EQUILIBRIUM  187 

of  the  territory  within  which  occurs  all  natural  action.  They 
are  dead-lines  which  not  only  matter,  but  even  thought,  may  not 
touch  without  ceasing  to  be.  A  great  deal  of  physics  may  be 
taught,  it  is  true,  without  mention  of  either  the  absolute  gas  or 
the  absolute  solid.  But  if  either  of  them  is  mentioned  at  all — 
and  their  use  is  common  in  teaching  even  elementary  physics — 
both  should  be  mentioned  together,  and  both  should  be  displayed 
as  supernatural  concepts.  Here  again,  neither  Siamese  twin 
should  be  presented  with  the  message  that  the  other  had  not 
yet  been  separated.  It  is  as  absurd  to  mention  to  the  student 
the  absolute  zero  of  temperature  without  parallel  mention  of  the 
perfect  gas,  or  vice  versa,  as  it  is  to  mention  the  force  of 
gravitation  without  its  inseparable  companion,  centrifugal  force, 
or  to  describe  a  chemist's  balance  as  a  pan  suspended  from  a 
rod  into  which  substance  may  be  put  for  weighing,  but  with  no 
mention  of  the  other  arm  and  pan  into  which  the  weights  are  put. 

Indeed,  the  very  description  of  matter,  to  the  student  fit  for 
generalities  at  all,  should  be  made  in  such  a  way  as  to  show 
that  solidity  and  expansive  fluidity  of  matter  are  purely  relative 
terms — that  all  matter  is  partly  solid  and  partly  gaseous,  that 
what  we  call  "solid"  matter  is  merely  matter  more  solid  than 
gaseous ;  that  what  we  call  "gas"  is  merely  matter  more  gaseous 
than  solid;  and  that  nowhere  in  nature  occurs  matter  which  is 
either  wholly  solid  or  wholly  gaseous. 

Although  water  alone  has  been  selected  for  illustration  in 
Fig.  12,  because  of  its  familiarity  in  solid,  liquid  and  gaseous 
states,  yet  these  general  conclusions  as  to  the  characteristics  of 
thermal  condition  and  action  of  mass  apply  equally  to  all  sorts 
of  matter.  All  that  is  needed  in  order  to  bring  the  thermal 
characteristics  of  any  substance  into  the  form  shown  is  to  alter 
suitably  the  scales  of  temperature  and  entropy.  Thus,  for  the 
hydrogen-oxygen  mixture  whose  curve  is  shown  as  hRGH,  all 
that  is  necessary  is  to  expand  sufficiently  the  temperature-scale 
and  reduce  the  entropy-scale,  and  its  field  of  fusion  and  vaporiza- 
tion would  appear  upon  the  diagram  in  much  the  same  form  as 
that  shown  for  water.  Similarly,  the  curves  for  any  of  the 
calcium  or  silicon  compounds,  which  remain  refractory  solids  at 
all  ordinary  temperatures,  might  be  brought  into  similitude  to  the 
water-steam  curve  merely  by  reducing  the  temperature-scale  and 
expanding  the  entropy-scale.  For  all  known  substances  pass 


188  ENERGY 

through  the  solid,  liquid  and  gaseous  states,  under  suitable  con- 
ditions. Fig.  12,  representing,  as  it  does,  not  only  these  three 
states,  but  the  processes  of  transition  between  them,  and  also 
chemical  transformation,  may  be  regarded  as  displaying  the 
thermal  action  of  all  known  substances — so  far  as  they  connect 
with  work  on  the  one  hand  and  with  chemical  energy  on  the 
other. 

With  this  in  mind,  let  the  curve  be  traversed  from  B  to  H,  as 
before,  but  with  thermal  conditions  treated  as  micro-mechanical 
ones. 

At  B  the  substance  is  a  "solid."  Energetically  speaking,  a 
solid  is  a  portion  of  mass  all  of  whose  parts  act  as  one,  in  their 
reaction  with  external  forces.  If  one  portion  moves,  all  the  rest 
move  simultaneously  an  equal  distance.  Between  the  particles  of 
the  system  there  is  no  motion  at  all,  except  tangential  motion. 
The  tangential  motion,  however,  must  be  plentiful. 

The  necessity  for  this  tangential  motion  can  best  be  under- 
stood by  reference  to  the  story  of  how  St.  Patrick  cleared  Ireland 
of  its  snakes:  Of  how  he  invited  all  the  toads  and  snakes  to  a 
banquet,  at  which  no  food  was  provided.  Thus  the  snakes  were 
led  to  devour  all  the  toads,  and  then  to  fight  and  devour  each 
other.  And  the  end  of  all  this  was  that  there  were  left,  finally, 
but  two  snakes,  containing  all  the  rest;  and  then  these  two  each 
got  the  other's,  tail  in  his  mouth  and  swallowed  and  swallowed, 
until  nothing  at  all  was  left ! 

For  the  force  of  gravitation,  it  is  plain,  if  left  to  itself,  would 
do  just  this.  It  would  deprive  matter  of  its  occupancy  of  space. 
For  we  know  of  no  such  thing  as  an  absolute  or  "perfect" 
density.  Every  dense  portion  of  matter  which  has  yet  been 
examined  has  proven  to  be  merely  an  association  of  portions 
still  more  dense,  the  further  characteristics  of  which  portions 
were  as  yet  obscure  to  us.  Mass  and  space  are  apparently  inde- 
pendent. The  mutual  gravitation  of  mass  increases  with  the 
square  of  the  propinquity,  with  no  known  limit.  Mass,  once 
drawn  into  propinquity  by  gravitation,  is  ever  urged  on  into  even 
greater  concentration. 

Here  is  obviously  unstable  equilibrium.  If  gravitation  were 
the  only  cosmic  force  the  universe  would  quickly  become  a  single 
geometric  point,  of  infinite  density.  The  universe  can  be  imag- 
ined as  continuously  existent  at  all  only  by  the  presence  of 


THERMAL  EQUILIBRIUM  189 

universal  tangential  motion,  developing  centrifugal  force  suffi- 
cient to  counterbalance  gravitation.  The  more  matter  concen- 
trates the  greater  must  be  these  tangential  velocities  and  cen- 
trifugal forces.  The  greater,  too,  must  be  the  rigidity  of  each 
tiny  whirling  mass-pair.  But  the  student  should  be  cautioned  at 
every  point  against  any  thought  of  any  absolute  or  final  limit 
to  the  smallness,  density,  speed  and  rigidity  of  such  "particles" 
of  matter.  He  should  be  reminded  that  it  is  just  as  foolish  and 
needless  to  speak  or  think  of  any  "ultimate"  or  "indivisible" 
particle  of  matter  as  it  would  be  to  speak — as  the  ancients  always 
did — of  any  solid  fundament  below  the  universe,  upon  which  it 
rested,  or  of  any  rigid  limits  to  space,  beyond  which  the  universe 
could  not  extend  and  where  existed — what?  Neither  beginnings 
nor  ends  of  anything,  whether  of  time,  space,  density,  solidity, 
elasticity,  intensity  of  energy  or  aught  else,  either  exist  in  nature 
or  can  be  comprehended  by  man. 

The  student  need  not  be  told  this.  But  his  teachers  very  much 
need  to  be  reminded  to  refrain  from  suggesting  to  him  the 
opposite  idea. 

Therefore,  to  return  to  the  cold  facts  of  Fig.  12,  however  far 
the  point  of  natural  condition  B  may  be  imagined  as  departing 
to  the  left,  approaching  the  condition  of  absolute  solidity  more 
and  more  slowly,  it  must  still  be  always  within  a  finite  distance  of 
its  mean  energetic  condition.  While  all  cold,  hard  solids  possess 
very  great  solidity,  rigidity,  density  and  inelasticity,  they  do  not 
altogether  lack  fluidity,  ductility,  space,  expansivity  and  elasticity. 
They  all  occur  at  very  low  temperatures,  relatively  to  their  liquids 
and  gases ;  but  they  always  possess  some  temperature.  Heat  can 
always  be  extracted  further  from  them.  In  short,  their  condition 
is  always  transmutable  into  liquids  and  gases  by  processes  of 
finite  dimensions.  We  therefore  surmise  that,  while  the  greater 
portion  of  the  mass  of  such  bodies  revolves,  in  exceedingly  dense 
particles,  with  almost  circular  motion  at  exceedingly  small  radius, 
and  with  very  great  rapidity  and  rigidity,  yet  there  exists  not 
only  some  slight  eccentricity  to  these  orbits,  but  a  minor  number 
of  mass-particles  revolve  about  highly  eccentric,  or  even  hyper- 
bolic, orbits.  The  tiny,  rigid,  tangential  pairs  arrange  themselves 
into  some  form  of  stable  equilibrium,  called  molecular  and  crys- 
talline. Between  and  about  and  out  from  them  shoot  the  minority 
of  satellitic  projectiles,  exerting  some  slight  expansive  vapor- 


190 


ENERGY 


pressure  and  affording  some  slight  degree  of  elasticity  and  some 
slight  manifestation  of  temperature.  At  human  temperatures 
many  crystalline  "solids"  exhibit  considerable  fluidity,  ductility, 
elasticity,  etc. ;  but  when  reduced  to  the  temperature  of  boiling 
hydrogen  these  same  substances  become  rigid  and  brittle  in  the 
extreme. 

It  is  such  "solids"  as  these  which,  by  their  relationship  in 
visible  distances  of  separation  and  visible  velocities  of  motion, 
embody  what  we  call  "mechanical"  energy.  Here  on  the  surface 
of  the  earth  we  see  these  solids  related  in  motions,  all  of  which 
are  below  the  lower  critical  velocity,  and  which  end  promptly 
in  collision  with  the  earth  and  with  each  other.  They  are 
therefore  inseparably  associated  with  impact  and  friction ;  that  is, 
with  thermogy.  Their  energetic  history  ends  always  in  this. 
Every  bit  of  mechanical  energy  aroused  here  on  the  earth's  sur- 
face, whether  by  our  artificial  heat-engines,  or  by  the  sun-heat 
acting  upon  wind  and  water,  or  by  the  moon  acting  upon  the 
oceanic  tides,  ends  its  existence,  in  a  surprisingly  short  time,  in 
transformation  into  heat.  The  region  of  mechanical  energy  is 
the  region  of  solidity,  opacity,  brittleness,  inelasticity  and  density. 


FAHRENHEIT  ZERO 


V 


OF  TEMPERATURE 


THE  REGION  OF  SOLIDITY,  OPACITY  AND  INELASTICITY;  OJ 
FRICTION,    IMPACT  AND   HEAT-ABSORPTION; 
WHERE  HEAT  WILL  NOT  TURN  INTO  WORK 
AND  WORK  INSISTS  UPON  TURNING 
INTO  HEAT. 

X-B 


ABSOLUTE  ZERO' 


PERFECT  SOLIDITY 


OF  TEMPERATURE 


FIG.  1 2 A. 


THERMAL   EQUILIBRIUM. 

It  is  the  region  of  impact,  friction  and 
thermogy.  There  all  "work"  insists  upon 
turning  into  heat,  and  heat  will  not  turn 
into  work.  It  is  the  region  displayed  at 
the  lower  left-hand  of  Fig.  12.  It  is  a 
region  which  belongs  especially  to  the 
solidified  planets  of  the  universe,  of 
which  our  earth  is  one.  It  is  particu- 
larly the  subject  of  what  we  call  the 
"applied  mechanics  of  engineering";  for 
while  engineering  does  not  deal  largely 
with  ice,  which  forms  the  illustration  in 
Fig.  12,  yet  it  does  deal,  almost  exclu- 
sively, with  substances,  such  as  iron, 
stone  and  wood,  which  are  as  far  below 


OF  THE 
^UNIVERSITY 


THE  MINIMUM  LIMIT  OF  TEMPERATURE 

FIG.  I2B. 


192  ENERGY 

their  mean  energetic  conditions  as  ice  is  below  boiling  water. 

Yet  the  prime  characteristic  of  this  region  is,  as  just  stated, 
the  thermogic  formation  of  heat.  Being  the  region  of  deficits 
of  heat,  entropy  and  temperature,  the  prime  result  of  its  activities 
is  to  make  good  these  deficits,  in  all  three  lines.  Its  activities 
being  of  an  extreme  energetic  nature,  they  show  every  disposition 
to  get  back  toward  the  mean  energetic  condition.  They  not  only 
tend  constantly  to  produce  entropy — which  inspection  of  Fig.  12 
shows  that  they  need  even  more  than  temperature — but  they 
produce  it  under  such  conditions  (namely,  a  deficit  of  internal 
expansive,  and  a  surplus  of  external,  pressure)  that  the  entropy- 
growth  is  promptly  converted  into  temperature-rise.  Whatever 
causes  may  be  offered  in  explanation  of  how  these  solid  bodies 
ever  got  into  this  extreme  condition  portrayed  at  B,  Fig.  12, 
there  can  be  no  doubt  as  to  their  unwavering  tendency  to  return 
toward  centrality,  along  the  path  BCADE. 

The  impact  and  friction  constantly  occurring  between  such 
solid  bodies  develops  entropy  in  collision,  as  has  been  described, 
by  smashing  them  into  finer  and  finer  fragments,  until  collision 
is  no  longer  possible.  Simultaneously,  to  these  fragments  are 
imparted  sufficient  velocities  of  tangential  motion  to  enable  them 
to  remain  separated  fragments,  without  falling  together  in  obe- 
dience to  mutual  gravitation.  This  process  we  call  the  creation 
of  heat.  So  long  as  the  substance  is  still  solid,  the  result  of  this 
increase  in  tangential  energy  is  to  arouse  a  resistance  from 
without,  which  compresses  the  widened  circular  orbits  until 
some  portion  of  the  energy  is  squeezed  into  a  radial,  or  satellitic, 
form.  This  increase  in  radial  energy  is  perceptible  from  with- 
out, whereas  the  tangential  energy  was  not,  and  leads  us  to 
observe  directly  that  the  "temperature"  of  the  body  has  risen. 
It  is  much  more  indirectly  and  slowly  that  we  have  come  to  a 
knowledge  of  the  latent,  or  tangential,  increase  in  entropy  which 
preceded  this. 

In  the  case  of  solids  below  the  fusion-point  the  internal 
tendency  to  increase  of  volume  and  fluidity,  with  increase  of 
entropy,  is  resisted  by  external  forces  which  we  are  forced  to 
call  "crystalline/'  in  lieu  of  fully  understanding  them.  In  the 
case  of  liquids  below  their  boiling-points,  however,  the  internal 
expansive  forces  are  resisted  by  an  external  fluid-pressure  which 
is  familiar  to  all;  and  the  action  of  this  external  pressure, 


THERMAL  EQUILIBRIUM  193 

in  squeezing  the  entropy-gains  into  temperature-gains,  is 
exactly  that  of  the  piston  of  the  air-compressor  or  the  cush- 
ioning steam-engine,  in  squeezing  volume  into  pressure  and 
temperature. 

When  such  points  as  C,  or  D,  or  F  are  reached,  however,  the 
effects  of  thermogy,  in  developing  entropy,  volume  and  elasticity, 
is  no  longer  opposed  effectively  by  the  external  pressure.  The 
processes  of  fusion,  vaporization  and  dissociation,  respectively, 
occur  in  purity,  as  developments  of  entropy,  latent  heat,  volume 
and  elasticity  at  constant  pressure — as  a  development  of  sub- 
division and  disgregation  of  matter,  and  of  radius  of  tangential 
motion,  without  any  increase  in  radial  kinetic  energy. 

It  thus  becomes  plain  how  matter  which  possesses  a  deficit  of 
radial  energy,  of  satellitic  mass,  of  volume,  expansive  pressure 
and  elasticity,  with  a  surplus  of  tangential  velocity,  density  and 
rigidity,  begets  naturally  those  things  which  it  lacks  and  rids 
itself  of  those  things  which  are  in  surfeit,  as  it  passes  along  its 
path  of  thermal  conditions,  BCADEFGH. 

It  is  now  to  be  noted  most  carefully  that  the  motive  power  of 
those  peculiarities,  which  started  it  along  this  path,  dies  out  as 
it  proceeds.  That  is  to  say,  as  it  gains  radial  energy,  tem- 
perature and  satellitic  mass,  and  more  particularly,  as  it  gains 
entropy,  volume  and  elasticity,  the  effectiveness  of  impact  and 
friction  for  heat-development  die  out.  The  causes  which  forced 
the  substance  toward  its  mean  energetic  condition  lose  their  force 
as  that  condition  is  reached,  and  counter-tendencies  begin  to 
prevail.  Thermal  or  other  energetic  conditions,  like  pendulum- 
bobs,  lose  their  motive  power  as  they  approach  their  central 
positions. 

This  does  not  occur  abruptly  or  completely.  Gradually 
rigidity,  density,  inelasticity  and  opacity  fade  away,  but  they 
never  completely  disappear.  We  know  of  no  substance  which 
does  not  possess  these  qualities  to  some  slight  degree,  which 
does  not  carry  on  some  slight  thermogic  action.  Even  the  most 
rarefied  of  gases  possess  some  slight  viscosity  and  absorption. 

With  these  diminutions  in  impact  and  friction  must  be  con- 
sidered also  thermal  conductivity  and  the  absorption  of  radia- 
tion. Of  these,  in  detail,  we  know  nothing,  except  that  their 
results  are  identical  with  those  of  impact  and  friction.  Here 
too,  in  the  progress  along  the  thermal  path,  the  likeness  again 


194  ENERGY 

appears.  Speaking  generally,  it  is  the  forms  of  matter  which 
are  most  rigid  and  dense,  and  most  subject  to  impact  and  fric- 
tion, which  possess  the  highest  rates  of  surface-absorption  of 
heat  and  its  conductivity  throughout  the  mass.  To  this  rule 
there  are  many  minor  exceptions,  but  they  are  local  and  insig- 
nificant. The  broad  fact  is  as  stated.  While  the  contrast 
between  solids  and  liquids  in  these  matters  is  not  marked,  that 
between  liquids  and  gases  is  so.  The  gases  are  most  difficult 
to  impart  heat  to.  Air  is  much  harder  to  heat  than  is  saturated 
or  slightly  superheated  steam,  and  steam  is  much  harder  than 
water.  As  to  solids,  the  difficulty  in  heating  them  does  not  lie 
in  any  lack  of  absorptivity,  but  in  bringing  ordinary  thermal 
media,  which  are  usually  liquid  or  solid  themselves,  into  effective 
contact.  Therefore,  as  matter  proceeds  along  the  thermal  path 
from  the  solid  extreme  toward  the  gaseous,  it  loses  first  its 
liability  to  impact  and  friction,  and  secondly  its  liability  to  the 
receipt  of  entropy  by  conduction  or  absorption. 

Moreover,  the  liability  to  thermogy  does  not  depend  solely 
upon  the  condition  of  the  body's  surface.  The  rate  of  heat- 
transfer  depends  also  upon  the  difference  in  temperature  between 
contributing  and  receiving  body.  When  the  recipient  is  a  very 
cold  solid,  nearly  all  other  bodies  are  warmer  than  it,  and  very 
much  warmer,  and  contribute  heat  rapidly.  But  as  the  substance 
grows  warmer  it  leaves  behind  it,  one  by  one,  those  neighbors 
which  were  recently  able  to  impart  heat  to  it,  and  in  turn  begins 
to  impart  heat  to  them.  Until  finally,  when  it  has  become 
unusually  hot,  as  toward  the  H-extreme  of  the  thermal  path,  it  is 
only  the  semi-occasional  body  which  is  still  warmer  than  it,  and 
so  able  to  contribute  heat  to  it. 

At  this  end  of  the  path,  too,  rigidity,  viscosity,  impact  and 
friction  have  almost  entirely  disappeared.  In  their  place  now 
appears  an  excess  of  volume,  expansive  pressure,  fluidity,  elas- 
ticity and  often  incandescence.  Here  work  will  no  longer  turn 
into  heat,  by  thermogy,  to  appreciable  degree.  Instead,  laborlty 
has  become  the  prevailing  phenomenon.  The  heat  insists  upon 
turning  into  work.  And  what  heat  cannot  find  conditions  suit- 
able for  its  conversion  into  work,  insists  upon  radiating  and 
conducting  itself  away  to  colder  bodies. 

Viewed  constructively,  the  molecule  now  appears  to  consist 
almost  entirely  of  minutely  subdivided  particles  moving  in 


THERMAL  EQUILIBRIUM  195 

highly  eccentric  and  chiefly  hyperbolic  orbit.  There  still  exist 
nuclei,  but  they  embody  a  minor  portion  of  the  mass.  There 
still  exists  tangential  motion,  but  it  is  now  an  insignificant  base 
for  the  prevailing  superpermanency  of  radial  energy. 

It  is  because  of  this  paucity  of  tangential  motion  that  it  is  so 
difficult  to  impart  further  heat  to  the  gas,  by  thermogy;  for 
impact,  friction,  conduction  and  the  absorption  of  radiation,  all 
must  occur  through  the  tangential  components  of  motion, 
although  they  may  find  final  expression  in  radial  motion.  It  is 
because  of  the  superabundance  of  radial  energy,  on  the  other 
hand,  that  it  is  so  easy  to  raise  the  temperature  of  a  gaseous 
body  by  compression;  for  compression  acts  directly  upon  the 
radial  component,  being  transformed  into  tangential  motion  only 
secondarily,  under  the  squeezing  of  the  nuclei  into  greater  pro- 
pinquity.  Only,  it  is  to  be  noted,  as  the  temperature  of  a  gas 
increases  so  also  does  its  pressure,  other  things  being  equal ;  and 
as  the  condition  of  the  substance  becomes  extreme  in  the  H- 
direction  it  becomes  increasingly  difficult  to  find  a  force  capable 
of  compressing  the  gas,  although  it  may  become  increasingly 
fluid  and  elastic  and  capable  of  compression  as  it  goes. 

Energetic  Gravitation.  There  thus  become  visible  in  the 
thermal  field  two  fundamental  gravitational  tendencies.  One  of 
these  is  for  all  substances,  and  particularly  those  to  the  left  of 
the  mean  energetic  condition,  to  pass  horizontally  to  the  right,  by 
thermogy.  The  other  is  for  all  substances,  but  particularly  those 
above  the  mean  energetic  condition,  to  pass  vertically  downward 
in  temperature,  by  labority.  Roughly  speaking — and  perhaps 
accurately  too — the  tendency  to  fall  in  vertical  intensity  (or,  in  this 
case,  in  temperature)  is  proportional  to  the  intensity  itself,  or 
the  distance  from  the  horizontal  axis  of  absolute  zero  of  tem- 
perature. In  saying  this  we  have  not  in  mind  the  force  with 
which  it  tends  to  fall,  but  the  chances  of  that  force  being  able 
to  prevail. 

Similarly,  the  tendency  of  energy  to  increase  in  quantity- 
factor  (in  this  case  entropy,  although  the  statement  applies 
broadly  to  the  mass-pairing  or  quantity-factor  of  any  form  of 
energy),  by  "collision,"  or  resistance  encountered  in  motion,  is 
proportional  to  the  distance  of  the  energetic  condition  from  the 
vertical  axis  of  absolute  zero  of  soliditv,  or  from  the  condition 
of  the  perfect  gas,  at  the  right.  In  this  we  see  that  we  have 


196  ENERGY 

unconsciously  formed  the  habit  of  speaking  of  entropy  with  the 
positive  and  negative  signs  reversed  from  what  they  naturally 
should  be.  Just  as,  in  speaking  of  space-energy,  in  order  to  be 
consistent  we  have  been  forced  to  substitute  the  idea  of  pro- 
pinquity, or  lack  of  space,  for  space  itself,  as  a  measure  of 
intensity,  so,  in  speaking  of  thermal  energy,  in  order  to  be 
consistent,  we  should  always  refer  to  entropic  changes  in  terms 
of  lack  of  solidity.  Solidity  tends  to  decrease,  in  the  universal 
thermogic  aspect  of  thermal  phenomena,  just  as  persistently  and 
uniformly  as  temperature  tends  to  decrease  in  their  work- 
performing  or  laborious  aspects.  Had  the  idea  of  entropy  and 
thermogic  tendencies  been  originally  thus  handed  out  to  the 
student  right  end  foremost,  as  a  simple  universal  tendency  of 
solidity  to  disappear  in  impact  and  friction,  quite  similar  to  the 
tendency  of  temperature  to  disappear  in  work-performance,  the 
subject  would  never  have  become  so  enshrouded  with  mystery 
and  awe  as  it  is  at  present.  But  now,  so  strong  is  habit,  it  will 
be  a  long  and  difficult  task  before  the  customary  algebraic  signs 
of  entropy  will  be  reversed. 

These  two  universal  tendencies  are  the  energetic  gravita- 
tions. Each  prevails  as  constantly  as  does  the  Newtonian  cen- 
tripetal gravitation  of  mass  and  centrifugal  gravitation  of  motion. 
In  mechanical  energy  these  tendencies  are,  as  to  its  intensity, 
that  of  either  unusual  velocity,  unusual  space  or  unusual  pro- 
pinquity to  decrease  to  a  minimum;  as  to  extensity,  that  of  the 
mass-pairing  of  matter  to  proceed  to  a  maximum,  in  its  further 
and  further  subdivision  (or,  in  other  words,  the  tendency  of  the 
solidity  of  all  matter  toward  a  minimum).  In  thermal  energy 
these  tendencies  are  that  of  temperature  to  decrease  and  that  of 
entropy  to  increase.  All  thermal  phenomena  are  the  result  of  a 
balance  between  these  two  gravitational  tendencies ;  for  the 
gravitations  act  always  in  opposition,  in  a  counterbalance  between 
each  other  and  with  outside  conditions.  Sometimes  one  prevails 
over  the  other,  sometimes  the  other  over  the  one ;  sometimes  both 
together  prevail  over  outside  forces,  sometimes  both  are  over- 
come thereby. 

If  Fig.  12  be  held  up  by  its  upper  left-hand  corner,  the 
thermal  path  BCADEFGH  will  then  appear  as  the  somewhat 
irregular  path  of  a  pendulum-bob,  swinging  from  the  point  of 
suspension.  Just  as  the  bob  of  a  real  pendulum  is  impelled 


THERMAL  EQUILIBRIUM  197 

toward  its  central  position  from  either  side,  whichever  it  may 
chance  to  reach,  so  is  the  thermal  condition  of  matter  impelled 
always  toward  its  mean  energetic  condition,  from  either  thermal 
extreme  which  it  may  chance  to  attain,  by  these  two  vast  gravita- 
tional tendencies.  Across  this  great  arc  of  physical  condition 
all  thermal  phenomena  are  constantly  swinging  back  and  forth, 
as  in  a  gigantic  pendulum.  For  the  ranges  of  the  arc  are  indeed 
gigantic.  From  the  coldest  density  of  solid  matter  lost  to  view 
in  interstellar  space,  on  the  one  hand,  to  the  highest  temperatures 
of  the  incandescent  suns  and  the  extreme  tenuity  of  the  luminous 
nebulae  and  comets'  tails,  on  the  other,  thermal  happenings  are 
constantly  swinging.  And  no  heat-action  on  the  surface  of  the 
earth  can  be  understood  without  reference  to  both  of  these 
extremes. 

What  combinations  of  conditions  in  nature's  vast  laboratory 
may  have  led  to  the  existence  of  suns  and  nebulae  on  the  one 
hand,  and  of  remote  dark  stars  and  cold  meteorites  on  the 
other,  is  a  great  and  open  question.  Its  answer  has  nothing 
to  do  with  the  present  argument.  The  fact  remains  unques- 
tionably fact  that,  these  extremes  of  heat-condition  once  in 
existence,  their  tendencies  must  be  as  described. 

Yet  the  conclusions  therefrom  must  not  be  too  hasty.  Carnot 
and  others  before  him  noticed  the  downward  tendency  of  tem- 
perature. Clausius  noted  the  outward  tendency  of  entropy — or 
the  downward  tendency  of  solidity.  It  was  Zeuner  who  deduced 
therefrom  the  quite  unwarranted  conclusion  that  the  entropy 
of  the  world  was  not  only  tending,  but  was  actually  moving, 
toward  a  maximum.  It  was  Lord  Kelvin  who  put  these  ideas 
together  into  the  doctrine  of  the  steady  "degradation"  and  loss 
of  availability  of  energy. 

These  mistaken  conclusions  depend  upon  seeing  these  two 
tendencies  as  working  always  and  everywhere  together  to  a  single 
end,  viz :  to  the  reduction  of  all  energy  to  heat,  and  all  heat  to  a 
maximum  of  entropy  and  a  zero  of  temperature-difference. 
The  facts  are  just  the  opposite  of  these.  The  two  tendencies 
are  always  opposed.  Entropy  cannot  be  increased  unduly  except 
by  processes  which  simultaneously  increase  the  intensity  of  tem- 
perature and  its  availability  for  labority.  Therefore  entropy 
itself  constitutes  a  factor  in  availability.  For  thermal  radiation 
and  work-performance  it  becomes  available  when  extreme  in 


198  ENERGY 

the  positive  direction.  For  thermogy  and  mechanical  energy  it 
becomes  available  when  in  an  extreme  condition  which  we,  mis- 
takenly, call  "negative"  (as  at  B,  Fig.  12).  For  matter  in  this 
extreme  condition  embodies  as  great  an  intensity,  or  availability, 
of  mechanical  energy  as  that  in  the  extreme  condition  at  H 
embodies  intensity  and  availability  of  thermal  energy.  Indeed, 
the  B  conditions  are  attained  by  matter  only  when  it  becomes 
so  remotely  isolated  from  its  fellows,  in  the  heavens,  that  any 
occurrence  of  thermogic  collision  at  all  must  develop  so  much 
heat  as  to  transfer  instantaneously  all  the  matter  involved  into 
an  equally  extreme  condition  of  temperature,  as  at  H. 

The  right  or  wrong  of  these  ultimate  conclusions  of  the 
physicist  is  of  little  importance  to  the  engineer;  but  a  correct 
understanding  of  the  familiar  thermal  phenomena  upon  which 
they  are  based  is  so.  It  is  therefore  worth  while  to  state  how 
the  doctrine  of  degradation — as  commonly  taught,  if  not  as  Lord 
Kelvin  stated  it — is  inconsistent  with  the  natural  facts. 

This  doctrine,  stated  briefly,  says  that  the  hot  half  of  the 
universe,  consisting  chiefly  of  the  suns,  is  steadily  becoming 
colder,  by  radiating  its  heat  to  the  colder  half,  which  latter  is 
steadily  becoming  warmer  by  absorbing  it.  Therefore  all  tem- 
perature-differences, and  with  them  all  availability  of  heat  for 
work,  must  ultimately  die  out.  The  fallacy  here  is  twofold,  and 
both  its  aspects  are  obvious  to  any  technical  graduate. 

In  the  first  place,  which  is  the  cold  half  of  the  universe? 
What  is  the  average  temperature  of  the  material  world?  Cer- 
tainly far  above  any  temperatures  familiar  upon  the  earth's 
surface.  The  mean  temperature  of  the  earth  itself,  considering 
its  interior,  must  be  above  2,000°  F.  Yet  it  is  a  cold  planet. 
The  planet  Jupiter,  exceeding  all  the  other  planets  together  in 
mass,  possesses  a  low  red  heat  even  at  its  surface.  The  sun,  of 
enormous  mass,  several  hundred  times  that  of  all  the  planets 
together,  possesses  a  mean  temperature  certainly  upwards  of 
10,000°  F.  Unquestionably  the  mean  temperature  of  all  the 
solid  matter  of  the  universe — using  the  term  solid  here  to  signify 
all  mass-portions  which  can  be  distinguished  as  separate  units, 
like  the  stars,  etc. — is  a  white  heat.  It  is  only  the  minute  frag- 
ments, scattered  occasionally  throughout  space,  remote  from  the 
incandescent  centers  of  congregated  mass,  or  suns,  which  are  ever 
"solid"  in  the  sense  which  we  use  in  engineering. 


THERMAL  EQUILIBRIUM  199 

Professor  Poynting  has  written  beautifully  concerning  the 
temperature-equilibrium  of  matter,  showing  how  the  external 
temperature  of  any  celestial  body  is  the  result  of  a  balance 
between  absorption  and  radiation.  He  shows  that  it  is  only  as 
we  inspect  the  smaller  and  smaller  bodies,  more  and  more  remote 
from  the  incandescent  centers,  that  the  lower  temperatures  are 
found.  The  temperature  of  solid  bodies  is  determined  almost 
solely  by  radiation — quite  as  it  is  with  the  articles  in  a  room 
containing  a  fire.  Those  nearest  the  fire  are  the  warmest,  and 
those  most  remote  are  the  coolest.  As  heat  radiates  from  the 
incandescent  suns  of  the  heavens  it  penetrates  regions  which 
become  colder  and  colder  as  it  goes.  Sooner  or  later  it  must 
all  be  intercepted  and  absorbed.  But  there  is  no  lower  limit  of 
temperature  beyond  which  radiation  will  not  extend,  provided 
it  chances  to  escape  absorption  long  enough.  It  is  only  the  mean 
distance  of  separation  between  solid  bodies  in  interstellar  space 
which  determines  the  mean  lower  limit  of  temperature  to  which 
radiation  falls,  before  it  experiences  arrest  and  absorption. 

Temperature,  then,  is  not  a  function  of  time,  except  in  a 
minor  fashion,  but  primarily  of  separation.  It  is  only  as  mass 
is  separated  into  remote  fragments  that  it  can  become  cold,  hard 
and  solid.  Time,  it  is  true, -introduces  a  lag  into  the  adjustment, 
so  that  departing  bodies  are  always  hotter  than  the  temperature 
of  equilibrium,  while  approaching  ones  are  colder;  but  this  is 
merely  incidental. 

The  earth,  then,  belongs  decidedly  to  the  cold  half  of  the 
universe.  If  so,  we  should  have  observed  here,  according  to  the 
degradation-theory,  a  steady  accumulation,  rather  than  degrada- 
tion, of  temperature.  Yet  it  is  the  obvious  opposite  of  this  which 
forms  the  main  support  for  the  degradation-hypothesis ! 

In  the  second  place,  the  doctrine  of  degradation  implies  the 
existence,  somewhere  in  the  universe,  of  an  immense  mass  of 
matter  which  has  somehow  become  cold  enough,  and  is  large 
enough,  to  be  capable  of  absorbing  all  the  radiation  from  the 
millions  of  suns  already  known  to  human  astronomy — not  to 
mention  the  fact  that  we  are  discovering  more  suns  every  day, 
almost  as  fast  as  we  can  count  and  catalog  them.  The  amount 
of  matter  requisite  to  do  this  must  be  enormous — far  greater 
than  the  mass  of  all  the  suns  together — and  it  must  be  extremely 
cold.  For  there  is  a  rigid  lower  limit  to  temperature-range,  the 


200  ENERGY 

absolute  zero,  so  that  matter  may  be  warmed  up  by  only  a 
thousand  degrees  or  so  before  it  loses  its  solidity ;  but  above 
there  is  an  unlimited  range  of  temperature  through  which  radiat- 
ing matter  may  cool  itself  down.  Any  engineer  who  has  handled 
steam-condensers  will  appreciate  the  task  of  finding  a  proper 
cooling-medium  for  absorbing  all  the  sun-heat  of  the  heavens, 
without  danger  of  overheating  the  medium. 

There  is  ample  interstellar  space  to  contain  all  this  matter- 
though  there  might  be  an  awkward  question  why  it  did  not 
interfere  more  obviously  with  light-transmission  through  space. 
But  the  unanswerable  question  is :  How  did  this  matter  ever  get 
so  cold?  It  could  not  have  done  it  by  expansive  labority ;  for 
that  is  a  process  confined  to  the  gaseous  substances,  and  could 
never  reduce  a  substance  to  a  solid,  nor  even  make  any  approach 
thereto.  Labority  tends  to  result  in  a  low-temperature  rarefied 
gas,  and  might  account  for  the  formation  of  cold  attenuated 
nebulae,  but  never  of  solid  planets. 

As  for  cooling  by  radiation,  or  thermogy,  it  is  inconceivable 
that  any  body  could  ever  so  cool  itself  in  that  way  as  to  turn 
around  again  and,  in  the  same  locality,  proceed  to  absorb  heat; 
and  that  at  enormous  rates,  in  enormous  quantities.  It  is  quite 
imaginable  that  a  body  should  so  cool  by  radiation  that  its  rate 
of  radiation  should  become  almost  zero.  The  cold  bodies  of 
remote  interstellar  space  are  in  this  condition.  But  it  is  incon- 
ceivable that  its  rate  should  ever  squarely  reach  zero,  and  still 
less  that  it  might  ever  develop  a  deficit  of  temperature,  against 
its  own  radiative  tendencies. 

The  fact  is  that  radiation  is  everywhere  cooling  matter. 
Thermal  energy  never  tends  in  any  other  direction  than  down- 
ward  in  temperature.  It  is  the  very  universality  of  radiation 
and  temperature-drop  in  heat  which  denies  the  doctrine  of 
degradation. 

Yet,  in  spite  of  this  universal  tendency  of  heat  down- 
temperature  and  the  resultant  inconceivably  vast  flood  of  radia- 
tion pouring  forth  from  every  sun  into  the  furthermost  crevices 
of  space,  the  mean  temperature  of  the  universe  remains  constant. 
Temperature  is  being  recovered  as  fast  as  it  is  lost.  Thermally, 
it  never  returns;  but  mechanically  it  does.  As  heat,  it  flows 
only  downward  in  temperature.  But  it  finds  chance  to  flow 
down-temperature  only  as  it  flows  outwardly  into  space;  and 


THERMAL  EQUILIBRIUM  201 

in  remote  space  it  can  find  embodiment  only  in  exceedingly 
remote,  rigid  and  inelastic  solids.  But  in  such  solids  is  also 
embodied  the  quintessence  of  mechanical  energy.  The  intensity 
and  availability  which  has  been  lost  to  thermal  forms  has  been 
regained  in  mechanical  form. 

Ultimately  these  remote,  cold,  hard  solids  must  end  their 
existence  in  collision,  resulting  in  gasification,  expansion  and 
incandescence.  The  swing  of  the  pendulum  will  have  been 
reversed.  The  intensity  and  availability  of  mechanical  energy 
which  they  embodied  will  have  been  lost,  and  in  its  place  will 
reappear  availability  for  radiation  and  elastic  work-performance. 

Thus  there  exist  always  and  everywhere  two  great  thermal 
tendencies,  which  are  balanced  against  each  other  in  cosmic 
equilibrium : 

First:  All  heat  tends  always  and  everywhere  to  fall  down- 
temperature,  either  by  radiation  or  by  work-performance. 
Nowhere  in  the  universe  has  ever  been  observed  any  cessation 
or  reversal  of  this  tendency.  Usually,  the  vast  flood  of  radiant 
energy  pouring  outward  into  space  from  all  the  countless 
millions  of  suns  finds  its  chance  for  direct  radiation  into  remote 
space.  This  is  true  of  the  bulk  of  all  sun-radiation;  and  in 
this  form  traveling  outwardly  at  the  inconceivable  rate  6f 
186,000  miles  per  second,  it  may  exist  for  eons.  Light  is  as 
permanent  a  form  of  energy  as  mechanical  energy  or  heat. 
The  universe  maintains  its  stock  of  it  as  permanently  as  it  does 
one  of  the  space  or  motion-energy  of  celestial  bodies.* 


*Of  the  radiance  emitted  by  our  own  sun,  for  instance,  about  one  hun- 
dred millionth  is  arrested  and  degraded  before  the  confines  of  its  own 
system  are  reached  and  passed.  Beyond  those  confines,  apparently,  is 
nothing  to  arrest  it  until  the  next  solar  system  is  reached.  When  that 
occurs,  assuming  the  similarity  of  all  solar  systems  to  our  own,  wherein 
about  one-fiftieth  of  all  the  surface  presented  is  dark,  about  one-fiftieth 
of  the  radiation  arrested  would  be  degraded,  by  collosion  with  opaque 
bodies.  The  other  forty-nine  fiftieths  would  be  arrested  by  the  central 
sun  and  reflected  without  degradation. 

But  even  these  proportions  are  divisions  of  what  is  itself  but  a  minute 
fraction  of  the  whole.  Assuming  that  the  mean  inter-stellar  distance  is 
but  eleven  light-years,  the  proportion  of  the  entire  radiation  arrested  by 
each  solar  system  would  be  only  a  decimal  fraction  of  the  whole  consist- 
ing of  unity  at  the  seventeenth  decimal  place.  In  other  words,  it  would 
not  be  until  100,000,000,000,000,000  solar  systems  had  been  met,  after 
II  X  io17  years  had  elapsed,  that  all  of  the  original  radiation  from  our 
sun  would  have  ceased  its  continuous  existence,  and  been  either  reflected 


202  ENERGY 

In  small  part,  however,  this  great  current  of  radiation  finds 
itself  intercepted  by  a  planet  equipped  with  air  and  water.  It 
becomes  entangled  in  the  tasks  of  raising  ocean-water  into 
clouds,  or  rearing  lofty  forest-trees  for  making  coal-beds,  or 
driving  reluctant  engine-pistons  before  it  as  it  goes.  Imprisoned 
in  the  latent  form  of  lifted  weights  or  stored  hydraulic  reser- 
voirs, or  of  chemical  energy  of  wood,  coal,  oil  or  gas,  it  may 
tarry  shortly  here  on  earth  before  it  proceeds.  But  only  briefly. 
Sooner  or  later  it  has  regained  its  thermal  form  and  liberty 
and  is  off  again,  outward  into  space  and  downward  in  tem- 
perature, never  to  return  voluntarily  an  inch  or  a  degree,  as 
heat.  Beaten  back  up-temperature  temporarily  it  may  be,  by 
superior  force,  as  in  our  air-compressors  and  refrigerating- 
machines;  but  it  always  resists  stubbornly  and  may  be  confined 
only  temporarily.  Soon  it  eludes  us  again,  and  is  off  for  the 
frightful  abysses  of  interstellar  cold  and  darkness — to  relieve 
them  as  it  may. 

Secondly:  All  lack  of  heat  (or  solidity)  tends  always  and 
everywhere  to  increase  in  entropy,  by  impact  or  friction,  or  to 
decrease  its  embodiment  in  solidity.  When  the  wandering  radia- 
tion finally  reaches  its  destination  and  is  absorbed  by  some 
remote  solid,  the  lower  the  temperature  of  its  new  home  the 
greater  must  be  the  latter's  separation  from  the  great  mass- 
center  which  did  the  radiating,  and  the  greater  must  be  its 
rigidity  and  inelasticity.  Always  and  everywhere  such  remote 
hard  solids  tend  to  fall  toward  the  greater  aggregations  of  mass. 
This  tendency  is  just  as  incessant  as  is  that  of  heat  downward 
in  temperature  and  outward  in  space.  So  remote  and  hard  are 
most  of  them,  and  so  intense  is  their  kinetic  energy  when  they 
do  fall,  that  they  are  fit  only  for  the  manufacture  of  incandescent 
suns.  But  some  are  quite  near  the  surface  of  the  earth,  stored 
in  the  hills,  and  upon  their  way  toward  greater  propinquity  they 
too  may  stop  a  moment  to  perform  work  for  us — driving  our 
water-wheels,  accelerating  our  railway-trains  on  down-grade,  or, 
as  in  the  case  of  the  moon,  cleansing  our  coasts  with  tidal  flow. 

But,  like  the  heat,  we  can  arrest  these  only  temporarily.    The 

or  degraded;  and  since,  at  each  star,  forty-nine  fiftieths  of  this  would  have 
been  reflected  without  degradation,  it  is  not  until  fifty  times  the  above 
number  of  solar  systems  should  have  been  met,  after  55  X  io18  years,  that 
all  of  the  original  radiation  would  have  suffered  true  degradation. 


THERMAL  EQUILIBRIUM  203 

tendency  is  all  in  one  direction,  and  is  irresistible.  The  sand 
washed  down  from  the  hills  never  returns.  Each  year  the  earth 
is  a  smaller  sphere  than  before,  and  a  colder,  harder  one. 
Nothing  will  ever  recover  this  ground  until  the  earth's  final 
collision  with  some  dark  star  ends  its  history  as  an  earth,  and 
begins  the  story  of  a  new  sun  and  solar  system  of  habitable 
planets. 

Equilibrium  between  Interchangeable  Forms  of  Energy. 

Thus,  just  as  it  has  been  found  that  within  mechanical  energy, 
and  so  also  within  heat,  if  heat  be  a  "mode  of  motion,"  there 
exists  an  eternal  equilibrium  between  spacial  and  kinetic  energies, 
so  exists  also  between  mechanical  and  thermal  energies  an  eternal 
equilibrium  of  two  great  opposed  tendencies,  or  gravitations. 
Heat,  as  heat,  tends  always  only  in  one  direction:  downward  in 
temperature,  turning  into  work  on  the  way,  if  it  must.  Work,  as 
work,  tends  always  only  in  one  direction :  downward  in  velocity, 
propinquity  and  solidity,  turning  into  heat  as  it  goes.  Tempo- 
rarily and  locally  either  tendency  may  overcome  the  other;  but 
universally  they  are  perfectly  balanced.  The  mean  availability 
of  each  form  of  energy  remains  eternally  constant. 

The  Rejuvenation  of  Intensities  of  Energy.  Thus,  while 
it  is  an  invariable  law  that  the  tendency  of  either  heat  or  mechan- 
ical energy  must  be  downward  in  intensity,  so  long  as  it  retains 
its  original  form,  yet  it  is  an  equally  invariable  law  that,  sooner 
or  later,  this  downward  tendency  must  result  in  transformation 
of  the  energy.  When  the  energy  takes  its  new  form  the  intensity 
also  takes  a  new  form.  Herein  appears  a  most  important  fact, 
viz :  the  degree  of  intensity  of  the  new  form  of  energy  is  inde- 
pendent of  that  of  the  old. 

Just  how  far  to  urge  the  accuracy  of  this  statement  the 
writer  is  uncertain.  It  might  almost  be  said  that,  as  far  as  the 
old  intensity  was  below  the  mean  energetic  condition,  the  new 
intensity  must  be  equally  above  mean  intensity.  Apparently  this 
is  the  only  true  and  consistent  statement;  but,  because  we  lack  a 
perfect  means  of  comparing  intensities  of  different  forms,  the 
writer  prefers  to  make  the  statement  tentatively,  until  he  can 
pursue  the  question  more  at  length. 

This  much  is  obvious,  however,  that  when  the  energy  has 
undergone  a  second  transformation,  back  into  its  original  form, 


204  ENERGY 

the  intensity  then  takes  a  form  which  can  be  compared  accu- 
rately with  its  degree  before  the  first  transformation.  It  is  now 
obvious  that  the  new  intensity  of  the  original  form,  rejuvenated 
by  having  undergone  a  double  transformation,  is  quite  inde- 
pendent of  the  original  intensity.  The  final  intensity  is  deter- 
mined only  by  external  conditions  which,  for  the  present,  may 
be  relegated  to  the  convenient  term  chance. 

This  fact  forms  the  basis  for  the  statement  of  a  general  law 
of  Gravitation  of  Intensities  of  Energy,  as  follows : 

Energy  tends  ever,  so  long  as  it  undergoes  no  transformation, 
to  gravitate  to  a  lower  degree  of  intensity.  This  tendency  ceases 
only  with  transformation.  The  energy  of  a  mass-system  can 
never  regain  intensity  except  by  two  methods:  (i)  by  a  con- 
tribution of  energy  from  some  external  mass-system,  or  (2)  by 
undergoing  a  double  energy-transformation,  into  some  other 
energy-form  and  back  again.* 

In  order  to  determine  the  extent  to  which  this  law  applies  to 
extensities  of  energy,  as  well  as  to  intensities,  we  should  have 
to  go  further  into  the  relationship  between  different  energy-forms 
than  seems  profitable  here.  To  show  where  this  question  leads, 
it  may  be  pointed  out  that  the  downward  gravitation  of  mechan- 
ical intensities,  tending  always  toward  collision,  impact,  friction 
and  thermogy,  is  equivalent  to  a  similar  gravitation  of  thermal 
tf.rtensity,  toward  an  increase  in  entropy,  or  a  decrease  in 
solidity.  This  fact  has  already  been  recognized  in  the  statement 
that  heat  possessed  two  gravitational  tendencies,  one  downward 
in  temperature  and  the  other  outward  in  entropy,  or  downward 
in  solidity.  It  now  becomes  clear  that,  of  the  two  energetic 
tendencies  possessed  by  each  form  of  energy,  one  is  identical 
with  one  of  the  two  belonging  to  a  strange  form  of  energy  on 
the  one  hand ;  the  other  is  identical  with  one  of  the  two  tendencies 
possessed  by  a  strange  form  on  the  other  hand.  To  trace  this 
interweaving  of  energy-forms  in  exact  language  is  beyond  possi- 
bility at  present;  but  this  correlation  of  intensities  of  different 
and  contrasted  energy-forms  must  be  mentioned,  as  essential  to 
the  principle  of  universal  energetic  equilibrium. 

This  law  the  writer  has  been  teaching  since  1899,  or  possibly  1898. 
In  1900  it  was  written  into  the  MS.  of  his  'Thermodynamics  of  Heat- 
engines,"  appearing  in  the  first  edition  of  that  book.  These  dates  are  from 
memory  only. 


THERMAL  EQUILIBRIUM  205 

The  Second  Law  of  Thermodynamics.  This  general  law 
of  energetic  gravitation  just  stated  has  been  represented,  in  the 
published  literature  of  thermodynamics,  by  the  so-called  "Second 
Law."  This  latter  confines  itself  solely  to  the  downward  tenden- 
cies of  temperature  alone.  The  present  aim  is  to  say  merely 
enough  to  show  that  the  law  regarding  temperature-fall  is  but  a 
narrow  and  special  expression  of  a  fundamental  energetic  prin- 
ciple, which  runs  through  all  known  forms  of  energy. 

It  should  also  be  added  that  the  absurdity  of  calling  this 
merely  thermal,  special  and  partial  aspect  of  a  great  principle  the 
"Second"  fundamental  law  has  now  become  apparent.  The 
engineering  world  is  now  far  too  frequently  occupied  with 
energy-transformations  to  countenance  longer  the  use  of  a  special 
nomenclature  for  each  different  form  of  energy,  independent  of 
and  inconsistent  with  all  the  others.  We  cannot  tolerate  one  set 
of  laws  for  mechanics,  another  for  heat,  a  third  for  chemical 
action  and  a  fourth  for  electricity,  when  all  four  of  these  sorts 
of  the  same  energy  are  at  work  in  nearly  every  engine-room  in 
the  country.  The  laws  of  energetics  must  be  codified  as  such, 
covering  all  special  forms  of  energy.  Should  convenience  then 
dictate  the  use  of  a  specialized  version  of  one  or  more  of  these 
laws,  in  any  particular  field  of  engineering,  it  would  be  well 
enough.  But  the  foundation  should  be  broad  and  secure. 

Fortunately  for  this,  the  order  of  importance  of  the  basic  laws 
of  energetics  appears  to  coincide  with  the  order  of  their  chrono- 
logical appearance.  The  Conservation  of  Mass  was  discovered 
first,  the  Conservation  of  Energy  second,  and  the  Conservation 
of  the  other  Factor  of  energy  than  mass — whether  it  be  called 
Motion,  Space,  Propinquity,  Temperature,  Intensity  or  what  you 
please — came  last.  This  apparently  reserves  the  place  of  Fourth, 
rather  than  ''Second,"  for  the  Law  of  Gravitation  of  Intensities, 
which  is  distinctly  a  secondary,  rather  than  a  primary,  principle. 

The  Summation  of  Energetic  Intensities.  The  true  ener- 
getic intensity  of  any  mass-system  is  in  reality  the  sum  of  the 
intensities  of  its  several  forms  of  energy.  Its  thermal  intensity, 
for  instance,  may  become  very  low  by  reason  of  its  mechanical 
intensity  becoming  very  high,  and  vice  versa.  But,  according  to 
the  law  of  the  conservation  of  energy,  the  sum  must  tend  to 
remain  constant.  Where  the  chances  of  environment  throw  the 
bulk  of  this  total  intensity  into  any  one  form,  there  may  appear 


206  ENERGY 

to  be  a  creation  or  a  loss  of  intensity;  but  it  is  only  apparent, 
not  real. 

Every  mass  large  enough  to  constitute  a  radiant  center  is 
constantly  attracting  to  itself  quantities  of  matter  so  cold  and 
solid  that  their  impact  contributes  enormous  funds  of  heat. 
Even  the  earth  is  said  to  collect  some  twenty  million  meteorites 
daily.  In  these  collisions  occurs  an  example  of  a  basic  ener- 
getic phenomenon,  the  summation  of  intensities.  The  sun,  for 
instance,  may  be  taken  as  the  hottest  known  body.  Yet  however 
hot  it  may  be,  it  can  never  be  hot  enough  to  prevent  a  body 
which  falls  into  it  from  increasing  its  energetic  intensity.  It 
may  be  so  hot,  it  is  true,  and  therefore  so  gaseous  and  elastic, 
that  the  falling  body  cannot  make  it  perceptibly  hotter.  But  it 
can  and  must  increase  its  gaseous  volume;  and  the  intensity  of 
spacial  energy  thus  stored  will  abide  potentially,  without  loss 
with  time,  to  make  good  the  temperature-drop  which  would 
otherwise  occur  as  radiation  proceeds.  Incandescent  radiation, 
like  every  other  energetic  process,  cannot  assume  a  too  great 
intensity  without  producing  conditions  which  limit  its  further 
increase. 

It  now  becomes  plain,  too,  why  our  heat-engines  and  other 
machines  always  have  so  poor  an  efficiency.  It  is  because  we 
are  confined,  on  the  surface  of  the  earth,  to  a  locality  peculiar 
in  being  below  the  mean  temperature  of  the  universe.  If  we 
could  only  build  our  machines  of  nebular  gases,  instead  of  from 
wood  and  steel,  and  could  jacket  them  with  incandescence,  we 
should  soon  cease  our  complaints  of  poor  mechanical  or  thermo- 
dynamic  efficiency.  Indeed,  if  we  were  such  salamanders  as  to 
be  able  to  live  where  such  procedure  were  natural,  we  should 
no  longer  prize  temperature  and  motion  as  we  do  now.  Instead 
of  struggling  ever  to  secitre  small  supplies  of  super-temperature, 
to  warm  our  bones  and  run  our  engines,  and  then  struggling 
further  to  convert  a  fraction  of  this  into  much  prized  motion — 
only  to  have  both  heat  and  motion  leak  away  promptly  into 
dissipation — we  should  then  seek  everywhere  for  that  rarest  of 
all  things:  a  chill  and  a  bit  of  solid  fixity.  Everywhere  would 
be  heat.  Everywhere  would  be  motion.  Flames,  whirlwinds 
and  hurricanes  of  gigantic  dimensions  would  overwhelm  us  at 
every  hand.  Only  rarely,  and  as  a  great  prize,  might  we  find  a 
morsel  of  peace  and  quiet  coolness,  as  a  firm  foundation  for  our 


THERMAL  EQUILIBRIUM  207 

salamandric  purposes.  But  in  another  instant  that  too  would 
be  melted,  vaporized  and  swept  away  from  our  grasp  into  dis- 
sipation, by  the  universal  surplus  of  heat  and  motion.  We 
should  then  appreciate  as  facts  what  now  our  wits  ought  to 
teach  us,  viz:  that  cold  is  just  as  much  a  promoter-of  cycles,  and 
is  just  as  valuable  to  nature,  as  is  heat;  that  solids  and  motion 
are  inimical  phenomena;  and  that  happiness  does  not  consist  in 
always  having  our  own  way. 

These  facts  we  appreciate  already,  in  a  vague,  empirical  way, 
if  we  do  not  teach  them.  A  machine  built  entirely  of  solids  we 
know  to  be  inefficient ;  so  we  insert  fluid  lubricating-oil  between 
the  impinging  solids.  Or  we  put  our  solids  afloat,  for  efficient 
motion,  and  use  an  ocean  of  sea-water  as  a  lubricant.  Best  of 
all,  we  already  appreciate  the  efficiency  of  utilizing  the  gaseous 
atmosphere  as  a  lubricant,  between  our  rapidly  flying  air-ships 
and  the  earth. 


CHAPTER  XVI. 

TRANSFORMATIONS  AND  CONSERVATIONS. 

When  attention  is  turned  to  other  forms  of  energy  than  work 
and  heat,  while  questions  as  to  their  exact  structure  become 
more  and  more  obscure  and  uncertain,  yet  their  mutual  identity 
with  heat  and  work,  in  all  of  their  fundamental  characteristics, 
becomes  more  clear.  While  in  the  case  of  work  it  proved  to  be 
possible  to  analyse  all  features  with  exactness  (except  that  no 
expression  for  the  fund  of  tangential  energy  could  be  found), 
when  heat  was  reached  it  became  necessary  to  deal  merely  with 
general  attributes,  averaging  for  numbers  of  component  mass- 
portions  too  great  for  individual  treatment,  and  reserving  elas- 
ticity of  definition  to  cover  conditions  impossible  of  definition. 

As  the  discussion  proceeds  from  work  and  heat  to  the  other 
forms  of  energy,  the  haziness  of  ideas  as  to  exact  structure — at 
least,  when  treated  by  the  writer — must  extend  rapidly.  It  is 
surprising,  indeed,  that  even  an  apparent  identity  may  be  dis- 
cerned. Yet  careful  examination  reveals  considerable  ground 
common  to  all  the  energy-forms. 

First,  all  the  known  forms  of  energy  are  mutually  trans- 
formable. Some  part  of  any  fund  of  any  form  of  energy  is 
always  capable,  under  favorable  conditions,  of  transformation 
into  any  other  form;  and  in  every  case  the  Conservation  of 
Energy  holds  true.  Thus,  if  to  the  list  of  energy-forms  already 
discussed  there  be  added  electricity,  radiant  energy  (light),  and 
biological  (animal  or  vegetable)  energy,  instances  of  their  mutual 
transformation  familiar  to  the  student  can  be  found  between 
each  two,  with  one  or  two  possible  exceptions.  This  leaves  out, 
among  the  familiar  energy-forms,  only  sound ;  and  the  amount 
of  energy  involved  in  most  audible  phenomena  is  too  small  for 
perception  after  transformation  into  the  other  forms  listed. 

208 


TRANSFORMATIONS  AND  CONSERVATIONS    209 


These  instances  of  mutual  transformation  might  be  listed  as 


follows : 

Thermal  to  Mechanical : 

Thermal  to  Chemical : 
Thermal  to  Electrical : 
Thermal  to  Radiant : 
Thermal  to  Biological : 

Mechanical  to  Thermal : 
Mechanical  to  Chemical 
Mechanical  to  Electrical 
Mechanical  to  Radiant : 
Mechanical  to  Biological 


Expansion  under  heat;  the  steam- 
engine. 

Dissociation ;  the  lime-kiln. 

The  electropile. 

Incandescence ;  all  flames  and  lamps. 

The  direct  effect  of  sun-heat  upon  veg- 
etable and  animal  life. 

Impact  and  friction ;  compression. 

Detonation. 

The  dynamo  or  glass-plate  machine. 

(None  known.) 

(None  known — unless  the  stimulative 
effect  of  the  slipper  or  the  shingle  be 
admitted  to  scientific  dignity.) 

The  chemical  fire-engine. 

Combustion. 

Primary  and  secondary  batteries,  dis- 
charging. 

Phosphorescence. 

The  consumption  of  animal  tissue. 

The  electric  motor. 

Electrical  resistance. 

Secondary  batteries  undergoing  charge. 

Crookes  tubes. 

Galvanism  in  medicine. 

Animal  activity. 

Animal  heat. 

The  accumulation  of  fat  and  tissue. 

Animal     phosphorescence;    the    glow- 
worm. 
Biological  to  Electrical :    Animal  electricity ;  the  electric  eel. 

This  array  of  energy-transformations  is  most  impressive  and 
significant.  While  there  are  two  combinations  in  the  above  list 
for  which  no  instance  is  known  to  the  writer,  and  while  sound 
and  magnetism  would  have  to  be  added  to  the  list  to  make  it 
complete  for  the  inanimate  energies  of  the  earth's  surface,  with 
celestial  and  sociological  energies  on  beyond  in  either  direction, 


Chemical  to  Mechanical 
Chemical  to  Thermal : 
Chemical  to  Electrical : 

Chemical  to  Radiant : 
Chemical  to  Biological : 
Electrical  to  Mechanical 
Electrical  to  Thermal : 
Electrical  to  Chemical : 
Electrical  to  Radiant : 
Electrical  to  Biological : 
Biological  to  Mechanical : 
Biological  to  Thermal : 
Biological  to  Chemical : 
Biological  to  Radiant : 


210  ENERGY 

yet  virtually  it  may  be  said  that  there  is  known  to  man  an 
instance  of  mutual  trans formability,  in  either  direction,  between 
every  two  known  forms  of  energy.  Some  of  these  phenomena 
are  rare  and  obscure  in  occurrence,  while  some  are  of  every-day 
familiarity.  All  in  all,  they  cover  an  exceedingly  diverse  field 
of  intricate  natural  action. 

It  is  next  to  be  noted  that  the  identity  of  heat  as  a  mode  of 
mechanical  motion  originated  from  and  rested  upon,  until 
recently,  nothing  more  than  this  mutual  trans  formability.  When 
the  mechanical  equivalent  of  heat  was  once  determined  exactly, 
the  identity  of  their  natures  was  considered  proven.  Yet  it  is 
merely  because  the  transformations  between  heat  and  work  are 
so  much  more  familiar  than  those  between  the  other  forms — or 
perhaps  because  they  are  the  most  familiar  of  those  capable  of 
exact  definition,  which  electrical,  radiant  and  biological  energy 
are  not — that  their  identity  was  foreshadowed  so  long  ago  and 
is  now  adopted  with  so  little  question.  It  was  more  than  two 
centuries  ago  that  Robert  Boyle  intimated  the  identity  between 
heat  and  work.  It  is  more  than  one  century  since  Count 
Rumford  proved  the  fact  qualitatively,  and  almost  quantitatively, 
upon  a  large  scale.  It  is  almost  half  a  century  since  Joule  proved 
it  quantitatively  and  exactly.  And  yet,  in  this  entire  history  of 
the  progress  of  the  idea  of  the  identity  between  heat  with  work, 
not  one  bit  of  evidence  has  been  adduced  which  is  more  con- 
clusive than  the  universal  mutual  transformability  of  the  two. 

But  a  belief  in  mutual  identity  thus  supported  must  extend 
as  far  as  does  mutual  transformability,  viz :  to  any  and  all  com- 
binations of  energy-forms.  Such  evidence  has  to-day  accumu- 
lated, for  chemical  and  electrical  energies,  far  beyond  that  exist- 
ent for  work  and  heat  in  either  Boyle's  or  Rumford's  time,  if 
not  also  in  Joule's.  Their  equivalence  has  been  brought  into  the 
circle  with  the  accuracy  which  was  attained  for  heat  only  in 
Joule's  hands.  As  to  radiant,  biological,  sociological  and  celestial 
energies,  no  units  for  their  quantitative  measurement  exist,  and 
therefore  coefficients  of  equivalence  are  of  course  impossible; 
but  the  instances  of  their  mutual  transformability,  apparently 
with  quantitative  conservation,  are  multiplying  daily  in  familiar 
experience. 

It  is  also  to  be  noted  that  even  those  coefficients  of  equiva- 
lence which  have  already  been  determined  rest  upon  transforma- 


TRANSFORMATIONS  AND  CONSERVATIONS    211 

tions  in  only  one  direction.  Rumford's  and  Joule's  work  was  all 
done  in  the  production  of  heat  from  work.  No  fixed  equivalence 
in  the  production  of  work  from  heat  has  ever  been  sought  or 
found.  The  applicability  of  Joule's  equivalent  to  work-perform- 
ance by  heat  is  pure  assumption,  checked  only  by  indirect  evi- 
dence as  to  the  correctness  of  Carnot's  law.  The  same  is  true 
of  thermo-electric  equivalence.  We  know  that  a  definite  amount 
of  electricity  will  produce  a  certain  amount  of  heat,  but  not  that 
an  equivalent  amount  of  heat  will  produce  its  proper  quota  of 
electricity. 

The  question  as  to  the  identity  of  all  these  various  forms  of 
energy,  as  all  being  modes  of  motion  and  forms  of  separation 
between  mass-particles,  and  as  all  being  amenable  to  the  laws  of 
energetics,  lies  therefore  in  this  state  of  settlement:  That  the 
evidence  in  favor  of  the  identity  of  work,  heat  and  chemical 
energy  is  so  overwhelmingly  great  that  no  one  to-day  dares  to 
deny  it,  or  to  suggest  an  adequate  substitute  hypothesis ;  yet  that 
there  exists,  in  the  face  of  an  only  slightly  less  abundance  of  the 
same  form  of  evidence,  an  astounding  reluctance  to  admit  the 
same  identity  between  other  forms. 

Of  these,  electricity  occupies  an  intermediate  position.  Static 
electricity  has  been  positively  identified  as  a  matter  of  mass ;  but 
concerning  kinetic  electricity  we  are  still  forced  to  rely  upon 
inference.  Light  and  magnetism  are  still  open  to  the  most  varied 
hypotheses. 

The  Universal  Intel-changeability  of  Energetic  Form. 
Yet  mass  is  merely  the  only  accurate  measure  for  quantity  of 
matter.  The  next  best  definition  of  mass  is  as  that  which  exerts 
force  or  stress,  or  exhibits  strain  under  stress.  Without  the 
concept  of  mass  the  words  ''strain"  or  "stress"  or  "wave"  or 
"pulse"  lose  all  significance.  It  therefore  seems  to  the  writer 
that  those  who  insist  upon  the  doctrine  that  the  ether,  for 
instance,  is  not  massive  are  just  as  reckless  with  language  as 
were  the  earlier  writers  upon  phlogiston  or  the  degradation  of 
energy.  What  a  "strain  or  pulse  in  the  ether"  may  be,  if  it  does 
not  signify  the  dislocation  or  acceleration  of  mass,  the  writer 
cannot  imagine.  The  writers  who  use  words  to  describe  etheric 
action  which  imply  naught  but  a  reference  to  familiar  mechanical 
(that  is,  massive)  phenomena,  and  who  yet  simultaneously  dis- 
claim any  such  parallelism,  cannot  appear  otherwise  than  as 


212  ENERGY 

tangling  themselves  in  words — albeit  as  artistically  as  a  Laocoon. 

Even  the  late  Lord  Kelvin,  in  his  1907  paper  before  the 
British  Association,  assumes  that  the  ether  has  no  mass.  Yet 
he  speaks  of  a  "pulse"  or  "disturbance"  of  the  ether,  and  then 
assumes  without  argument  that  this  disturbance  involves  energy. 
But  if  the  ether  is  massless  its  disturbance  would  not  necessarily, 
nor  even  probably,  involve  energy.  The  words  "disturbance"  or 
"wave"  carry  an  energetic  significance  only  when  the  disturbance 
is  that  of  mass.  For  the  only  known  thing  which  absorbs  energy 
in  its  disturbance  is  mass.  The  ether  can  form  no  exception  to 
this  statement,  for  the  ether  is  unknown. 

Again,  he  says :  "There  is  no  difficulty  in  this  conception  of 
an  utterly  homogeneous  elastic  solid"  (the  ether).  There  is  no 
difficulty  in  the  concept  of  the  ether  as  a  solid ;  but  there  is  as  a 
homogeneous  elastic  solid.  The  only  energetic  systems  man  has 
been  able  to  dissect,  with  mathematical  accuracy,  are  the  celestial 
systems;  and  in  these  the  only  instance  of  either  elasticity  or 
energy  occurs  as  a  function  of  heterogeneity — of  a  dissociation 
and  interaction  at  a  distance  between  two  or  more  bodies  which 
may  be  quite  dissimilar,  and  each  of  which  may  be  totally 
inelastic,  but  both  of  which  are  massive  and  are  in  motion.  This 
elastic  action  at  a  distance  is  quite  compatible  with  the  "solidity" 
of  the  pair,  by  any  criterion  for  solidity  except  homogeneity. 
The  "homogeneous  elastic  solid"  is  much  like  Voltaire's  famous 
"Holy  Roman  Empire,"  which  turned  out  to  be,  upon  inspection, 
neither  holy,  nor  Roman,  nor  an  empire.  That  is  to  say,  we 
can  have  no  concept  of  a  truly  homogeneous  substance  as  pos- 
sessing any  qualities  whatever;  but  the  one  especially  complete 
lack-of-quality  which  a  homogeneous  body  must  have  is  absolute 
inelasticity  and  passivity. 

The  same  looseness  of  thought  and  diction  prevails  in  the 
frequent  reference  to  an  "indivisible  unit"  of  matter.  For  a 
thousand  centuries  man  has  existed  in  ignorance  of  any  portion 
of  matter  smaller  than  the  tangible  or  visible.  During  the  last 
century  of  this  thousand  he  has  confidently  regarded  the  chemical 
atom  as  the  ultimate  indivisible  unit — although  not  a  bit  of  the 
evidence  upholding  the  atomic  theory  indicated  that  the  atom, 
however  small  a  division  of  matter,  was  the  ultimate  subdivision. 
Nevertheless,  during  the  last  thousand  days  or  so  he  has  readily 
accepted  the  idea  of  the  electron,  a  thousand  times  smaller  than 


TRANSFORMATIONS  AND  CONSERVATIONS    213 

the  atom.  Yet  he  now  stands,  apparently,  just  as  firmly  as  ever 
for  the  idea  that  the  electron  is  now  really  the  ultimate  unit  of 
matter — although  he  already  knows  of  three  different  sorts  of 
electron,  implying  an  internal  configuration,  and  consequently  a 
relation  of  parts. 

The  sensible  man  of  the  educational  profession,  and  the 
sensible  undergraduates  as  well,  resent  these  childish  incon- 
sistencies. They  know  well  that  another  thousand  days  may  see 
the  proof  that  the  electron  is  itself  divisible  into  very  many  still 
smaller  mass-units,  as  yet  undiscerned.  They  can  see,  both 
rationally  and  instinctively,  that,  whatever  may  be  the  last  dis- 
covered smallest  portion  of  mass,  its  energetic  activity  must 
imply  at  least  some  degree  of  elasticity,  and  therefore  of  hetero- 
geneity of  structure,  held  apart  kinetically.  They  know  that  the 
one  thing  unknown  to  nature  or  the  laboratory  is  homogeneity; 
that  everything  proves,  upon  dissection,  to  consist  of  a  mere 
heterogeneous  relationship,  between  parts  which  individually 
possess  none  of  the  features  evinced  by  the  relationship.  The 
fact  that  many  things,  perhaps  familiar  things,  yet  remain  undis- 
sected  is  no  evidence  whatever  in  negation  of  this  idea. 

For  this  great  question  of  the  identity  between  the  several 
forms  of  energy  no  conclusive  evidence  can  be  expected.  There 
can  be  brought  to  bear  upon  it,  however,  a  second  line  of  indirect 
evidence.  This  rests  upon  the  identity  of  all  these  diverse 
energy-forms  in  their  constitution  and  their  characteristics  of 
action. 

The  Universal  Dualism  of  Dimension  in  Energetics. 
Early  in  the  study  of  mechanical  energy  it  was  pointed  out 
that  this  best  known  of  all  energy-forms  is  composed  of  two 
factors,  intensity  and  extensity.  These  two  factors  combine, 
not  as  a  sum,  but  as  a  product.  Neither  factor  may  vary  toward 
zero  except  as  the  other  factor  varies  toward  infinity.  There- 
fore, in  spite  of  the  unceasing  gravitation  of  each  factor  toward 
its  own  zero,  neither  may  travel  in  that  direction  with  other  than 
negative  acceleration  and  increasing  resistance,  because  it  is 
thereby  forcing  the  other  away  from  its  zero  with  positive 
acceleration. 

These  same  characteristics  apply  to  all  the  more  obscure 
forms  of  energy,  with  fair  completeness.  Each  form  of  energy 


214 


ENERGY 


possesses  a  dual  nature,  consisting  of  two  dimensions  or  factors, 
its  own  particular  forms  of  intensity  and  extensity  respectively. 
The  gravitational  tendency  of  each  of  these  factors  may  be 
observed. 

This    duality    of  nature  and    identity    between    the    several 
factors  may  be  indicated  by  the  following  table  :* 


FORM  OP 
ENERGY: 

FACTOR  OF  INTENSITY  : 

FACTOR  OF  EXTENSITY  : 

Name 

Unit 

Name 

Unit 

MECHANICAL: 
Potential: 
Approx.  : 
Exact: 

Kinetic: 
Approx.  : 

Exact: 

ELECTRICAL: 
Potential: 
Kinetic: 
CHEMICAL: 
Potential: 
THERMAL: 

Kinetic: 
Potential: 

Distance 
Propinquity 

/                       \ 

Feet 

Force 
Mass-squared 

Mass 
(Mass)2 

Charge 
Current 

Mass 

Entropy 
Entropy 

Pounds 
c  (Lbs.-*-GN2 

Lbs.-r-G 
(Lbs.-f-G)2 

Coulomb 
Ampere 

Molecular  wt. 
B.  t.  u. 

Feet-per-sec. 
(Feet-per-sec.)2 

^       distance  } 

Velocity 
(Velocity)2 

Total  mass 

Potential 
Potential 

Lbs.-5-G 

Volt 
Volt 

Temperature 
Disgregation 

Degree  (abs.) 
«         « 

Abs.  Temp. 
it 

Although  this  table  is  not  complete,  the  correspondence 
between  the  several  forms  of  energy,  in  the  possession  of  two 
factors  or  dimensions  of  energy,  one  of  which  is  probably  a 
function  of  motion,  distance  or  force  and  the  other  of  mass  and 
its  subdivision,  is  obvious.  In  mechanical  energy  the  items  are 
complete;  except  that  we  lack  a  unit  of  measurement,  or  even 
the  familiar  concept,  of  propinquity  as  the  intensity-factor  of 
potential  energy;  and  we  lack  an  exact  expression  for  the  tan- 
gential or  latent  fund  of  mechanical  energy. 


*The  writer  wishes  to  record  here  the  statement,  although  it  is  impossi- 
ble to  support  it  here,  that  what  study  he  has  made  of  social  and  eco- 
nomic energies  reveals  the  same  duality  of  form  running  through  them 
all.  Some  of  the  material  for  this  appears  in  his  "Cost  of  Competition." 
Much  yet  awaits  development.  Of  the  duality  of  factors,  however,  there 
can  no  longer  be  any  doubt. 


TRANSFORMATIONS  AND  CONSERVATIONS    215 

In  electrical  energy  all  the  items  are  complete;  but  only  in 
the  case  of  static  electricity  has  it  been  settled  that  the  extensity- 
factor  is  identical  with  mass.  The  intensity  of  electricity  classes 
itself  with  the  other  intensity-factors  only  by  its  forcefulness, 
with  its  gravitational  tendency  to  fall ;  for  electrical  action  always 
takes  place  away  from  the  locality  or  condition  of  higher  voltage 
toward  that  of  lower  voltage. 

In  chemical  energy  the  intensity-factor  is  wanting,  as  yet. 
The  only  definite  accomplishment  in  the  line  of  identification  is 
that  of  the  direction  of  spontaneous  chemical  action  as  always 
coincident  with  increase  in  entropy.  In  this,  chemical  action  is 
plainly  in  parallel  with  kinetic  mechanical  energy.  But  the  bare 
existence  of  a  duality  of  factors  is  as  apparent  in  chemical 
energy  as  in  the  other  forms.  The  mass  factor  is  stated  above 
as  merely  mass,  whereas,  to  accord  with  mechanical  energy,  it 
should  be  the  square  of  the  mass.  But  here  it  must  be  remem- 
bered that  chemical  measurements  concern  themselves  only  with 
varying  numbers  of  tiny  mass-systems,  each  called  a  molecule, 
embodying  energy  in  some  particular  chemical  form  of  arrange- 
ment. Apparently  the  mass,  energy  and  extent  of  mass-pairing 
of  each  molecule  are  the  same;  the  quantity-measurements  have 
to  do  only  with  the  number  of  molecules.  In  this  case  the 
quantity-factor  must  vary,  not  as  mass-squared,  but  as  mass. 

In  thermal  energy  the  two  factors  have  already  been  suffi- 
ciently discussed. 

With  the  other  forms  of  energy  the  identification  of  the  two 
factors  is  still  more  obscure.  Still,  so  far  as  our  knowledge  of 
these  other  energy-forms  goes,  it  falls  in  line  with  the  general 
concepts  which  were  based  upon  mechanical  energy  in  the  earlier 
chapters.  Thus,  we  know  comparatively  little  about  the  mechan- 
ism of  the  sound-wave  in  air;  yet  we  know  that  these  waves 
produce  pressure,  due  to  the  arrest  of  moving  particles  of  mass. 
In  the  case  of  the  radiant  energy  of  light,  also,  the  pressure 
developed  by  reflection  has  been  observed  by  several  independent 
observers. 

Even  in  the  field  of  the  most  recently  developed  and  least 
known  of  all  the  sciences,  that  of  radioactivity,  what  little  we 
know  falls  into  line  with  these  same  fundamental  concepts  of 
energetic  action.  In  radioactivity  there  are  plainly  two  variables, 
the  intensity  of  radiation  and  the  mass.  The  extreme  degree  of 


216  ENERGY 

concentration  itself,  of  energy  within  a  given  mass,  is  made 
more  comprehensible  by  our  earlier  conclusions,  drawn  from 
mechanical  energy,  as  to  the  unlimited  ability  of  mass  to  em- 
body energy. 

Again,  the  intensity  of  radioactivity  exhibits  the  most  obvious 
downward  tendency.  One  of  the  first  tasks  in  identifying  each 
newly  discovered  radioactive  substance  is  to  determine  the  time- 
rate  at  which  the  intensity  drops.  No  such  substance  has  been 
discovered  in  which  the  rate  of  radiation  increased  with  time. 

Again,  the  intensity  of  all  forms  of  radioaction  decreases, 
with  time,  by  what  is  known  as  the  half-rate  law.  That  is  to  say, 
its  intensity  decreases  by  one-half  in  equal  portions  of  time.  If 
the  rate  be  called  unity  at  any  instant,  and  the  period  of  time 
be  observed  until  that  rate  shall  have  fallen  to  one-half,  then  at 
the  end  of  the  second  equal  period  of  time  the  rate  will  have 
fallen  to  one-quarter,  at  the  end  of  the  third  period  to  one- 
eighth,  and  so  on.  If  these  rates  should  be  plotted  upon  rec- 
tangular coordinates,  with  time  for  the  other  axis,  there  would 
result  a  curve  of  the  form  of  an  equilateral  hyperbola,  asymp- 
totic to  both  axes.  In  other  words,  the  rate  of  radiation  could 
never  fall  to  zero,  no  matter  how  long  it  should  continue  to 
radiate;  and,  going  backwards  previously  to  the  time  of  first 
observation,  it  may  be  said  that  the  radiation  must  have  begun 
within  some  fairly  definite  recent  period,  before  which  no  radia- 
tion could  have  occurred  without  being  infinite  in  its  rate.  In 
this  the  variation  of  radioactive  intensity  with  time  is  quite 
similar  to  the  form  of  variation,  in  hyperbolic  function,  of  every 
other  energetic  function  which  has  yet  been  examined  in  this 
series  of  papers. 

Again,  many  of  the  manifestations  of  radioactivity  are  based 
apparently  upon  the  linear  motion  of  very  small  particles  of 
mass.  As  the  velocity  of  these  particles  approaches  that  of  light, 
186,000  miles  per  second,  their  mass  apparently  increases  very 
rapidly.  This  is  explained  as  due  to  the  mass  of  the  ether  which 
must  be  displaced  in  their  passage — just  as  the  inertia  of  a  ship 
is  really  that  of  the  hull  moving  forward  plus  that  of  the  water 
moving  astern  to  make  good  its  displacement.  This  explanation 
tacitly  admits  the  massiveness  of  the  ether.  But  even  if  this 
question  be  not  entered,  here  is  another  energetic  function — that 
between  the  apparent  or  effective  mass  of  the  moving  particle 


TRANSFORMATIONS  AND  CONSERVATIONS    217 

and  the  difference  between  its  velocity  and  that  of  light — which 
is  hyperbolic  in  form.  For  apparently,  if  the  velocity  of  light 
could  ever  be  attained  by  these  particles,  their  mass  would  have 
become  infinite ;  while  for  velocities  far  below  that  their  apparent 
mass  is  very  small. 

Again,  the  radioactive  substances  tend  to  degrade,  with  time, 
into  substances  more  stable  chemically.  Both  lead  and  copper 
are  thought  to  have  been  produced  from  radium  in  this  way. 
This  variation  of  chemical  mass-pairing  and  intensity,  within  a 
chemical  mass  which  remains  unchanged  in  the  aggregate,  is  an 
apparent  parallel  with  the  way  in  which  thermal  mass-pairing,  or 
entropy,  varies  within  a  mechanical  aggregation  of  mass  which 
itself  remains  unchanged. 

Energy-transformation.  In  all  of  these  dual  forms  of 
energy,  with  factors  varying  in  generally  hyperbolic  function 
asymptotic  to  two  limiting  zero-axes,  the  variation  of  each  factor 
occurs  in  counterbalance  against  the  other  in  stable  equilibrium. 
Given  sufficiently  favorable  conditions,  this  smooth  fluctuation  in 
stable  equilibrium  may  cover  an  unlimited  range.  There  is  no 
known  limit  to  the  upward  expansion  of  the  intensity-factor,  or 
the  outward  extension  of  the  extensity-factor,  except  external 
conditions. 

These,  in  every  natural  case,  dictate  a  certain  limit  to  the 
exaggeration  of  either  energy-factor  beyond  a  certain  point. 
The  hyperbolic  energy- function  here  becomes  discontinuous. 
The  energetic  equilibrium,  previously  stable,  becomes  unstable. 
Energy-transformation  sets  in.  The  appearance  of  the  energy 
to  human  senses  is  abruptly  altered.  The  gravitational  law  pre- 
viously prevailing  becomes  invalid.  It  is  only  with  especial  care 
that  the  continuous  force  of  even  the  laws  of  conservation — of 
mass,  energy  and  intensity — may  be  discerned. 

In  mechanical  energy  these  limits  of  stable  equilibrium  were 
fully  discussed,  and  their  results,  in  the  form  of  either  collision 
or  dissociation,  were  identified.  In  thermal  energy  these  same 
limits  are  visible,  but  their  exact  nature  must  be  inferred  rather 
than  observed.  Of  these  the  most  familiar  are  the  temperature- 
limits  at  which  occur  fusion,  vaporization  and  chemical  dissocia- 
tion. But  these  are  wholly,  or  largely,  internal  in  their  equilib- 
rium. Fusion  is  almost  independent  of  pressure;  vaporization 
is  balanced  against  mechanical  pressure;  dissociation  is  balanced 


218  ENERGY 

against  chemical  pressure.  In  addition,  thermal  intensity  is  in 
equilibrium  with  mechanical  pressure,  determining  whether  work 
is  to  be  performed  by  expansion  or  not.  It  is  also  in  equilibrium, 
in  the  thermopile,  with  electrical  intensity  and  resistance,  deter- 
mining whether  heat  shall  be  altered  into  electrical  energy  or  not. 
In  solutions,  temperature  is  in  equilibrium  with  the  proportionate 
presence  of  dissolved  and  undissolved  substance.  In  ignitable 
explosives  temperature  is  in  equilibrium  with  the  rigidity  or 
strength  of  the  chemical  structure  of  the  atom,  so  that  when 
more  than  a  certain  amount  of  thermal  energy  is  introduced,  per 
unit  of  mass,  the  instability  of  equilibrium  appears  most  spectacu- 
larly, in  violence  of  explosion.  In  the  detonating  explosives  is 
exhibited  a  similar  equilibrium  between  mechanical  intensity  and 
chemical  resistance.  Dynamite  may  be  heated  to  any  degree 
whatever,  so  that  it  ignites  and  burns ;  yet  its  chemical  structure 
is  stable  (except  for  combustive  action)  in  the  face  of  this 
thermal  intensity.  But  intensity  of  mechanical  shock  it  cannot 
withstand,  beyond  a  limited  degree. 

Indeed,  the  familiar  phenomenon  of  ignition  is  one  of  the 
best  illustrations  of  energetic  equilibrium.  Up  to  the  ignition- 
point  a  combustible  substance  absorbs  heat  in  the  same  way  as 
ice  does — with  gradual  change  in  entropy  and  temperature,  in 
stable  equilibrium.  But  as  the  ignition-temperature  is  reached 
the  chemical  equilibrium  becomes  unstable,  with  a  resultant 
transformation  even  more  striking  than  when  ice  melts. 

But,  since  it  has  been  said  that  all  these  illustrations  of  the 
more  obscure  energy-forms  are  explicable  in  terms  of  mechanical 
energy,  the  writer  searched  long  for  an  instance  of  mechanical 
energy-transformation  occurring,  as  the  result  of  too  great  con- 
centration and  intensity  of  energy,  in  some  more  familiar  mechan- 
ical fashion  than  celestial  collisions.  It  was  finally  found,  one 
summer's  day,  while  watching  a  steamboat  describe  a  long  circle 
across  a  still,  deep  harbor,  under  momentum  only.  The  water 
was  glassy  calm.  The  bow-waves  formed  themselves,  on  the 
side  of  the  boat  toward  the  center  of  curvature,  into  a  wide  uni- 
form arc,  which  moved  smoothly  and  noiselessly  across  the 
water's  surface,  narrowing  concentrically  as  it  came.  The 
intensity  of  motion-energy  in  the  water-waves  was  plainly  being 
increased,  by  the  concentric  direction  of  the  waves ;  yet  here  was 
no  disturbing  question  of  external  conditions,  as  when  the  wind 


TRANSFORMATIONS  AND  CONSERVATIONS    219 

drives  a  wave  until  it  breaks,  or  a  solid  beach  interferes  with  its 
progress  and  transforms  its  energy  into  heat. 

If  our  ideas  as  to  instability  of  equilibrium  resultant  from 
undue  intensity  of  energy  were  correct,  something  ought  to 
happen  here  soon;  and  it  did.  Finally,  almost  simultaneously 
around  the  extensive  arc,  the  waves  broke  noisily  into  foam, 
although  the  water  was  deep  and  the  air  was  still,  from  sheer 
overconcentration  of  energy.  A  large  portion  of  their  energy 
suddenly  became  heat  and  sound. 

Stability  in  Energetic  Transformations.  Yet  the  insta- 
bility of  equilibrium  visible  in  energy-transformation  is  itself 
limited  in  scope.  A  pendulum  with  the  bob  held  vertically  above 
the  point  of  support  is  in  unstable  equilibrium.  Released,  it  will 
transform  its  store  of  energy  abruptly,  with  positive  acceleration. 
Yet  the  result  of  this  action  is  to  remove  the  phenomenon  from 
the  field  of  instability,  into  one  where  the  equilibrium  is  stable. 

It  is  so  with  all  energy-transformation.  While  energy- 
transformation  is  initiated  only  when  the  equilibrium  is  unstable, 
yet  it  occurs  always  in  the  direction  of  recovery  of  stability. 

In  mechanical  energy  it  was  noted  that  too  great  intensity  in 
the  form  of  propinquity  begets  collision  and  energy-transforma- 
tion; but  that  the  result  of  this  is  a  form  of  energy,  heat,  in 
which  occurs  no  collision.  Too  great  an  intensity  of  velocity,  on 
the  other  hand,  begets  dissociation ;  but  the  result  of  dissociation 
is  to  transfer  the  mass-portions  into  such  propinquity  to  other, 
larger  systems  that  they  are  trapped  there  and  can  no  longer 
dissociate. 

Similarly  with  heat,  unusual  temperature  begets  energy- 
transformation  in  the  form  of  work-performance;  but  the  first 
result  of  work-performance  is  to  lower  the  temperature  and 
stop  the  transformation.  Unusual  lack  of  temperature  begets 
thermogic  energy-transformation;  and  the  first  result  of  this  is 
to  develop  entropy,  volume  and  elasticity,  so  that  the  thermogy 
is  retarded. 

The  same  is  true  of  chemical  energy,  as  familiarly  visible  in 
combustion.  A  combustible  mixture  of  gases,  if  ignited,  does 
not  burn  completely.  Combustion  is  retarded  by  two  things 
resultant  from  combustion:  (i)  temperature,  causing  disso- 
ciative tendencies  which  countervail  the  mutual  attraction  between 
fuel  and  oxygen;  and  (2)  chemical  pressure,  due  to  the  presence 


220  ENERGY 

of  a  preponderating  proportion  of  the  stable  chemical  products 
of  oxidation.  Both  resistances  to  further  combustion  rest  upon 
forms  of  thermochemical  equilibrium. 

Indeed,  we  are  told  that  in  such  a  mixture  of  gases  there 
never  exists  a  purity  either  of  separation  before  combustion  or 
of  combination  afterwards.  That  is  to  say,  in  every  combustible 
mixture  there  exist  before  combustion  some  small  portions  of 
the  products  of  combustion,  in  proportions  determined  by  molec- 
ular equilibrium  at  low  temperature.  When  the  mixture  is 
ignited  most  of  the  fuel  and  oxygen  combine,  but  not  all.  A 
portion  still  remains  in  dissociation,  held  apart  by  the  new  condi- 
tion of  thermal,  chemical  and  mechanical  molecular  equilibrium, 
determined  by  the  new  temperature  and  pressure  prevailing.  For 
each  different  temperature  and  pressure  there  is  a  different  pro- 
portion of  fuel  or  oxygen  still  uncombined.  Often  in  the  arts, 
as  in  gas-enginery,  this  proportion  is  sufficient  to  be  of  economic 
importance.  More  often  it  is  so  small  as  to  be  more  of  a  chem- 
ical curiosity.  But  always  it  is  there.  Apparently,  no  exaggera- 
tion of  condition  will  either  get  all  the  fuel  and  oxygen  apart,  or 
induce  them  wholly  to  combine. 

In  every  phase  of  natural  action  this  universal  equilibrium  is 
to  be  traced.  The  hot  summer-day  begets  transformation  of 
heat  into  electricity,  breeds  a  thunder-storm  and  furnishes  a  most 
spectacular  display  of  energy-transformation  in  unstable  equilib- 
rium. But  the  thunder-storm  "cools  the  air"  and  recovers  the 
weather's  equilibrium ;  and  no  more  storms  occur  until  heat  again 
accumulates  unusually. 

An  electric  current  finds  opportunity  to  enter  a  motor,  finds 
there  conditions  favorable  for  transformation  into  motion — that 
is,  an  unusual  intensity  of  field,  coupled  with  unusual  quantity 
of  current  across  it.  Motion  ensues.  But  the  immediate  effect 
of  the  motion  is  to  beget  a  counter-electromotive  force,  which 
so  reduces  the  current  that  the  transformation  is  reduced  from  a 
condition  of  unstable  to  one  of  stable  equilibrium. 

A  stream  of  water,  flowing  across  a  meadow,  washes  easily 
the  soil  from  the  banks  and  carries  it  with  it.  But  the  washing 
is  done  in  unstable  equilibrium.  On  whichever  side  it  occurs,  the 
water  is  thrown  in  that  direction,  by  centrifugal  force,  and  exag- 
gerates the  departure  from  a  straight  line  of  flow.  But,  as  these 
departures  on  either  hand  become  extreme,  the  length  of  the 


TRANSFORMATIONS  AND  CONSERVATIONS    221 

water-course  becomes  exaggerated  thereby.  The  hydraulic 
"slope"  of  the  stream  becomes  lessened,  and  its  velocity  of  flow 
too  low  to  carry  longer  an  appreciable  mass  of  suspended  earth. 
So  the  stream  adopts  finally  a  series  of  S-shaped  meanderings — 
in  which  the  disposition  to  pick  up  soil  is  stably  balanced  against 
the  disposition  to  drop  it — as  the  form  of  water-course  of  per- 
manent equilibrium. 

In  all  departments  of  nature  its  every  and  most  diverse 
aspect  must  be  understood,  first  of  all,  as  being  the  natural  and 
inevitable  result  of  preceding  causes,  acting  always  in  stable 
equilibrium.  The  forms  of  not  only  earth,  sea  and  solar  system, 
but  also  those  of  vegetable  and  animal  life,  can  be  nothing  else 
than  the  fruit  of  energetic  evolution,  reacting  with  former  self 
and  present  environment  in  an  eternal  stability  of  equilibrium. 
The  known  forms  have  survived  because  they  are  those  embody- 
ing stability  of  equilibrium. 

In  every  field  of  activity  known  to  man,  in  the  energetics  of 
moons,  molecules  and  men  themselves — in  individual  human  life, 
in  economics,  in  politics,  in  war  and  peace — the  continued  preva- 
lence of  stable  equilibrium  and  apparent  quiet  begets  an  accumu- 
lation of  intensity  which  periodically  surpasses  the  critical  limit 
and  begets  instability.  Spectacular  transformation  of  energy 
ensues.  But  the  instability  is  always  temporary ;  the  transforma- 
tion always  occurs  in  the  direction  of  recovery  of  stability. 
Everything  works  in  the  direction  of  its  own  demise  and  the 
birth  of  a  new  regime.  Natural  phenomena  never  progress 
smoothly  and  continuously.  In  nature  as  in  human  history,  in 
molecular  as  in  military  affairs,  in  the  celestial  chariots  of 
Phoebus  and  Aurora  as  in  a  modern  automobile,  matters  get 
ahead  by  a  series  of  explosions,  followed  by  relaxations  into 
lassitude. 

It  is  the  explosions,  not  the  intervening  periods  of  recupera- 
tion, which  we  perceive  and  by  which  we  characterize  the  energy- 
form.  To  only  one  out  of  a  hundred  does  "the  French  nation," 
for  instance,  mean  aught  or  more  than  a  Reign  of  Terror,  a 
Waterloo  and  a  Commune.  The  continuous  daily  life  of  the 
French  people  counts  for  nothing.  Yet  it  is  this  continuous  daily 
life  which  accumulates  the  energy  become  spectacular  in  the 
revolutions. 

It  is  thus  that  we  must  abandon  hope  of  securing  satisfactory 


222  ENERGY 

names  or  attributes  for  any  one  form  of  energy.  It  is  only 
transformations  of  energy  which  appeal  to  us.  We  know  nothing 
about  heat  as  heat.  It  is  only  as  it  enters  or  leaves  its  cryptic 
ant-hill  that  we  see  it.  When  it  transforms  itself  into  nervous 
shock  in  our  bodies,  or  into  volume,  pressure  or  electric  current 
in  our  thermometers,  or  into  work  in  our  engines,  or  into  light 
in  our  lamps,  we  say :  "Lo,  here  is  heat !"  But  in  none  of  these 
cases  is  it  the  heat  itself  which  we  perceive. 

The  Fundament  of  the  Energetic  Universe.  It  has  been 
a  constant  care,  in  the  preceding  chapters,  to  dislodge  the  pre- 
vailing concept  of  energy  as  a  flat-footed,  static  thing,  resting 
upon  an  absolute  zero  of  something  as  a  supporting  base,  and 
rising  therefrom  in  a  simple,  additive  way.  For  this  idea  of 
energy — of  all  energy-forms,  as  well  as  for  mechanical  energy — a 
concept  radically  different,  in  two  respects,  is  necessary.  First, 
all  energy-quantities  vary  on  either  side  of  a  mean  energetic 
value,  which  mean  condition  is  itself  unsupported.  Secondly,  the 
path  of  motion,  in  kinetics,  as  one  of  these  variables,  ranges  thus, 
in  eccentricity  of  conic-section  orbit,  on  either  side  of  the 
parabola,  as  the  mean  energetic  path.  The  parabola  is  the  funda- 
mental orbit,  the  sole  natural  geometric  base  for  all  things,  the 
orbit  of  unit  eccentricity,  embodying  equal  quantities  of  radial 
and  tangential  motion.  The  straight  line  has  no  place  in  ener- 
getics. 

This  needed  metamorphosis  of  our  ideas  is  so  great  as  to 
appear  impossible.  Yet  a  quite  similar  transfiguration  had  to 
be,  and  was,  accomplished  in  another  science — astronomical 
kinematics — as  much  as  three  centuries  ago.  In  the  days  when 
Vasco  di  Gama,  Christopher  Columbus,  Magellan  and  Drake 
were  opening  the  far  seas  to  European  commerce,  and  revolu- 
tionizing the  world's  ideas  as  to  the  nature  and  extent  of  its 
own  civilization,  what  was  the  astronomical  concept  upon  which 
rested  their  aids  to  navigation?  The  pre-Pythagorean  or  post- 
Ptolemaic,  of  a  flat  earth  which  served  as  a  fundament  for  the 
heavens,  relatively  to  which  zero-plane  of  reference  the  sun 
"rose"  and  "set."  By  the  time  of  Columbus  the  idea  of  the  flat 
earth  had  given  way  before  his  own  genius;  but  the  earth  still 
remained  as  the  center  of  interstellar  space.  And  with  the  mass 
of  people  the  idea  of  the  flat  earth  continued  tenaciously.  It  did 
not  disappear  from  all  of  our  Protestant  church-creeds  until 


TRANSFORMATIONS  AND  CONSERVATIONS    223 

within  the  last  half -century.  So  prominent  and  able  a  man  as 
President  Kruger,  of  the  Transvaal,  still  held  to  the  "simple" 
faith  in  the  flat  earth,  the  supported  skies  and  the  moving  sun,  as 
the  twentieth  century  dawned — although  the  intricacies  and 
inconsistencies  into  which  it  leads  are  beyond  bare  statement  here. 

Columbus  never  met,  in  all  his  stormy  voyages  in  tiny,  top- 
heavy  craft,  natural  obstacles  to  progress  so  great  as  he  every- 
where encountered,  in  public  opinion,  in  the  prevalence  of  these 
crude  ideas  as  to  the  astronomical  nature  of  the  universe.  For 
generations  navigation  suffered  untold  loss  because  of  public 
bigotry  in  refusing  to  countenance  true  astronomy.  Even  a 
century  later  than  Columbus  the  Gallilean  and  Copernican  philoso- 
phies almost  carried  their  advocates  to  the  stake;  and  although 
the  few  then  began  to  see,  the  rest  followed  slowly.  The  entire 
present  wealth  of  these  United  States  would  not  make  good  the 
losses  to  commerce  and  civilization  which  have  been  involved  in 
the  slow  reluctance  of  mankind  to  abandon  its  reliance  upon  a 
rigid,  tangible  support  for  the  heavenly  bodies  from  an  absolute 
base,  in  favor  of  a  faith  in  intangible  "action  at  a  distance,"  and 
that  too  about  an  unsupported  center,  as  a  sufficient  explanation 
of  celestial  mechanics. 

Yet  this  early  astronomy  was  no  more  crude — in  comparison 
with  the  Gallilean-Copernican  concept  of  a  central  sun,  itself 
moving,  unsupported,  through  space,  with  the  earth  and  planets 
revolving  about  it — than  is  the  "absolute  zero,"  "up-and-down," 
rectilinear  concept  of  energy,  as  an  attribute  of  homogeneous, 
indivisible,  ultimate  matter,  which  holds  sway  to-day,  when  com- 
pared with  the  truth.  Many  of  the  statements  now  taught  to 
our  youth  as  fundamental  principles  of  mechanical  science  are 
the  exact  reverse  of  the  truth. 

Yet  the  time  of  reform  is  now  upon  us.  Some  vital  change 
looms  imminent.  Energy  is  now  as  important  a  topic  as  navigation 
was  then.  The  great  industrial  and  monetary  interests  are  now 
linked  with  the  use  of  natural  energy  in  manufacture,  and  with 
the  manufacture  and  sale  of  energy  itself,  as  they  were  then 
with  transoceanic  discovery  and  commerce.  Just  as  navigation 
was  then  the  prime  factor  in  gigantic  transformations  in  human 
thought  and  political  institutions,  so  is  discovery  in  the  field  of 
energetics  now  the  guiding  cause  in  enormous  recent  and  immi- 
nent changes  in  public  opinion  and  democratic  institutions.  It 


224  ENERGY 

was  steam-transportation,  the  cotton-gin  and  the  telegraph  which 
fifty  years  ago  made  of  slavery — an  institution  which  had  existed 
beneficently  to  man  since  the  dawn  of  history — an  anachronism 
so  inefficient  and  disturbing  that  its  abolition  was  forced  upon 
the  nation,  upon  civilization,  at  whatever  cost  in  men  and  money. 
The  similar  or  greater  changes  in  the  form  of  our  social  organiza- 
tion which  now  promise  to  be  forced  upon  us,  as  the  inevitable 
result  of  the  more  recent  discoveries  of  the  telephone  and  trolley, 
the  gas-engine  and  the  steam-turbine,  hydro-electric  transmission- 
systems  for  light  and  power,  wireless  telegraphy  and  rural  free 
delivery,  are  yet  to  be  measured  out  in  nature's  laboratory. 
For  their  thorough  comprehension  and  their  safe  guidance  it  is 
imperative  that  the  run  of  practical  men  of  affairs  should  possess 
accurate  concepts  of  the  internal  energetic  action  and  possibilities 
of  large  and  intricately  organized  masses,  whether  of  molecules 
or  of  men.  But  before  we  may  hope  to  step  into  such  a  true 
comprehension  of  the  energetic  universe,  purified  from  its  present 
chaotic  mixture  of  inconsistency  with  complexity,  we  must  alter 
our  point  of  view  from  its  present  post-Ptolemaic  to  a  more 
Copernican  position.  We  must  get  off  the  surface  of  the  earth 
and  rise  above  every-day  human  standards,  before  we  may  grasp 
the  significance  and  the  majesty  of  that  every-day  phenomenon : 
energy-transformation.  Universal  law  holds  true  here,  as  else- 
where ;  but  we,  with  our  little  factories  and  heat-engines,  are  not 
the  fundament,  nor  even  the  center,  of  the  universe. 

It  may  be  true,  as  says  the  writer  on  astronomy  in  the 
Encyclopedia  Britannica,  that  the  Copernican  feat  of  removing 
the  center  of  the  celestial  system  from  the  earth  to  the  sun,  with 
its  immediate  unfolding  of  the  complex  mystery  of  the  planetary 
system  into  rational  simplicity,  accomplished  no  perceptible 
advance  in  the  science.  It  may  be  true  that  the  future  of  physics 
lies  solely  "in  the  sixth  decimal  place."  The  writer  does  not 
believe  either  statement.  The  sort  of  astronomy  which  knows 
nothing  outside  of  the  sixth  decimal  place  possibly  was  not 
advanced  by  Copernicus.  But  the  sort  of  astronomy  which  could 
never  have  been  revealed  by  sixteen  decimal  places,  applied  to 
the  old  ways — the  sort  of  astronomy  which  fires  men's  minds 
with  new  ideals  and  devotions,  which  tears  inside  out  old  world- 
systems  of  bigoted  faith  and  cruel  superstition — this  sort  could 
never  have  lived  without  Copernicus  and  Gallileo. 


TRANSFORMATIONS  AND  CONSERVATIONS    225 

But  even  the  coldest  and  most  mathematical  science  pro- 
gresses thus.  The  re-definition  of  terms,  the  codification  of  laws 
and  the  projection  of  rational  hypotheses  are  all  as  powerful  aids 
to  efficient  observation  as  is  the  latter  to  the  accurate  growth  of 
theory.  And  just  at  present — particularly  in  both  the  engineering 
and  the  economic  fields — the  empirical  side  is  unquestionably 
overpulling  on  the  whirfie-tree ;  we  possess  far  more  data  than 
we  have  yet  properly  digested. 

The  Unity  of  All  Energetic  Action.  There  can  be  little 
question  that  the  present  broad  trend  of  scientific  progress  is 
along  the  lines  of  an  accumulation  of  complexity  of  detailed  data, 
but  with  a  simultaneous  precipitation  therefrom  of  an  increas- 
ingly simple,  consistent  and  unified  set  of  underlying  principles. 
This  is  true  not  only  in  pure  science,  but  also  in  engineering  and 
the  other  applied  sciences — and,  above  all,  in  sociology,  the  most 
important  of  all  sciences  to  human  happiness.  The  problem  has 
nowhere  been  better  stated  than  by  Sir  William  Ramsay,  in  his 
1904  address  before  the  St.  Louis  meeting  of  the  International 
Congress  of  Arts  and  Sciences,  in  discussing  the  imminent  prob- 
lems of  chemical  science : 

"I  have  already,  in  an  address  to  the  German  Association  at 
Cassel,  given  an  outline  of  the  grand  problem  which  awaits 
solution.  It  can  be  stated  shortly,  then:  While  the  factors  of 
kinetic  and  gravitational  energy,  velocity  and  momentum,  on  the 
one  hand,  and  force  and  distance  on  the  other,  are  simply  related 
to  each  other,  the  capacity  factors  of  other  sorts  of  energy — • 
surface,  in  the  case  of  surface-energy;  volume,  in  the  case  of 
volume  energy ;  entropy,  for  heat ;  electric  capacity,  when  electric 
charges  are  being  conveyed  by  means  of  ions;  atomic  weight, 
when  chemical  energy  is  being  gained  or  lost — all  these  are 
simply  connected  with  the  fundamental  chemical  capacity,  atomic 
weight,  or  mass.  The  periodic  arrangement  is  an  attempt  to 
bring  the  two  sets  of  capacity-factors  into  a  simple  relation  to 
each  other ;  and  while  the  attempt  is  in  so  far  a  success,  inasmuch 
as  it  is  evident  that  some  law  is  indicated,  the  divergences  are 
such  as  to  show  that  finality  has  not  been  attained.  The  central 
problem  in  inorganic  chemistry  is  to  answer  the  question,  Why 
this  incomplete  concordance?" 

But  is  it  a  fact,  as  Sir  William  states,  that  the  factors  of 
mechanical  energy  are  so  simply  related?  Is  it  not  true  that 


226  ENERGY 

j  other  sciences  are  obscure  chiefly  because  our  mechanical  con- 
cepts are  confused,  vague  and  often  inconsistent?  Is  it  not 
lively  that,  when  we  have  swept  our  eyes  clear  of  cobwebs  in 
regarding  our  more  familiar  forms  of  energy,  the  more  obscure 
ones  may  stand  out  in  much  better  definition?  At  any  rate,  to 
do  anything,  however  imperfect,  toward  the  improvement  of  our 
scientific  basis  for  this  broader  aspect  of  all  the  natural  sciences, 
as  mere  departments  of  a  single,  consistent  whole,  is  the  highest 
aim  to  which  human  thought  may  now  aspire. 

Indeed,  this  is  the  basic  object  of  all  true  education — as  dis- 
tinguished from  mere  training — to  open  the  eyes  to  the  invisible, 
to  broaden  the  narrowness  of  view  of  ignorance.  For  this  there 
is  needed  only  an  early  inculcation  of  the  unity  of  all  nature. 
We  do  not  hesitate  to  place  early  in  the  high-school  course  the 
doctrine  of  the  Conservation  of  Matter,  in  spite  of  the  infinite 
variety  of  form  in  which  matter  appears.  We  regard  the  doc- 
i  trine  of  the  Conservation  of  Energy,  throughout  similar  diversity 
of  form,  as  the  core  of  our  college-taught  science.  Why  should 
not  the  parallel  doctrines  of  the  Conservation  of  Intensity,  of 
the  Duality  of  Energy-factors,  and  of  the  unity  of  all  extent- 
factors  with  mass-subdivision,  be  taught  as  equally  basic  concepts  ? 
To  many  writers,  too,  the  assumption  seems  to  come  nat- 
urally that  the  different  localities  and  scientific  departments  of 
universal  action  are  quite  independent  of  each  other,  or  even 
discordant.  The  fact  of  unity,  interdependence  and  identity 
seems  to  call  for  some  rigid  proof,  before  it  can  be  accepted. 
They  seem  to  forget  that  basic  principles  are  always  axiomatic. 
They  seem  to  forget  that  a  "proof"  is  nothing  more  than  the 
dependence  of  a  conclusion  upon  its  premises,  and  not  possibly 
of  greater  import  than  those  premises.  But  to  the  writer  the 
identity  of  all  sorts  of  natural  action  lies  in  the  axiomatic 
premises.  It  is  their  discordance  which  must  be  proven.  Since 
the  discovery  of  universal  gravitation  and  the  speed  of  trans- 
mission of  light  the  universe  has  been  unified  over  distances 
hopelessly  beyond  human  comprehension,  by  bonds  measurable, 
as  to  time,  in  terms  of  human  life  and  action.  Since  the  dis- 
covery of  the  mutual  interchangeability  of  light  with  heat, 
motion,  animal  and  vegetable  life  and  inanimate  electricity,  since 
the  invention  of  the  spectroscope  and  the  bolometer,  the  unity 
and  identity  of  natural  action  in  the  most  remote  abysses  of 


TRANSFORMATIONS  AND  CONSERVATIONS    227 

interstellar  space  with  the  most  familiar  of  every-day  happenings 
here  upon  the  surface  of  the  earth  have  become,  as  was  said, 
axiomatic.  Their  underlying  principles  are  not  merely  similar; 
they  are  palpably  identical.  The  burden  which  lies  upon  us  is 
not  that  of  proving  that  they  are  identical.  It  is,  rather,  to  define 
in  detail  what  are  their  differences. 

Nor  does  this  idea  of  unity  mean  that  all  forms  of  energy  are 
but  allotropic  forms  of  one  basic  form,  whether  that  be  electrical 
or  mechanical  or  chemical.  It  means  that  each  is  a  different 
outward  aspect  of  a  single  hidden  inner  nature,  which  latter  we 
may  never  hope  to  comprehend.  To  the  writer,  mechanical  is 
the  most  familiar  form  of  energy;  therefore  he  naturally  refers 
all  other  extents  of  energy  to  mechanical  pairs  of  mass,  and 
all  other  intensities  to  visible  space  and  motion.  Yet  he  does 
not  know  what  either  mass,  or  space,  or  motion  really  is,  and 
has  no  expectation  or  desire  that  any  one  will  ever  know. 
Similarly,  to  Professor  J.  J.  Thomson,  for  instance,  electrical 
is  the  most  familiar  form  of  energy;  so  he  naturally  refers  all 
other  extents  of  energy  to  electrical  charge  as  a  base.  Yet  he 
makes  no  pretence,  I  believe,  that  the  true  inner  nature  of  the 
electrical  charge  will  ever  be  known,  however  minutely  we  may 
dissect  it  further  in  the  future.  To  Professor  Ramsay,  again., 
chemical  energy  is  the  natural  base.  Yet  here  again  is  no  better 
hope  of  ultimate  comprehension.  Mere  reduction  into  terms  of 
something  else  is  all  that  science  may  ever  attempt.  The  unity, 
but  not  the  ultimacy,  of  nature  is  the  lesson  of  science. 

"Energy,"  then,  is  a  dual  circular  chain  of  links.  Each 
"form"  of  energy  constitutes  a  link  in  the  circle.  As  we  walk 
about  this  circle  we  may  regard  the  different  links  lying  nearest 
us,  the  chemical  or  thermal  or  mechanical  as  may  be.  From 
these  combined  impressions  we  judge  the  inward  nature  as  best 
we  may;  just  as  we  know  an  actor  only  after  seeing  him  in 
many  parts,  under  diverse  make-ups.  But  none  may  say  that 
any  one  of  these  make-ups  is  the  actor  himself. 

In  this  great  movement  of  human  thought  and  action,  due  to- 
civilization's  current  change  of  front  from  its  earlier  material- 
spiritual  basis  to  its  present  ultra-energetic  aspect,  it  is  as 
natural  that  the  engineer  should  forge  to  the  front,  as  the  first 
to  understand  and  to  do,  as  it  was  in  earlier  times  for  soldiers, 


228  ENERGY 

navigators  and  lawyers  to  be  the  leaders  of  men.  But,  if  he  is 
to  rise  to  his  opportunity  in  the  new  century,  the  engineer  must 
be  more  broadly  equipped.  He  must  understand  not  only 
machines  and  individual  men,  but  vast  masses  of  men — not 
hundreds  or  thousands  of  them,  but  millions  and  tens  of  millions 
of  them.  As  his  first  start  toward  equipment  for  his  public  and 
private  duty  he  must  grasp  the  great,  fundamental  principles  of 
all  energetic  action.  He  must  understand,  as  well  as  memorize, 
these  three  basic  laws  of  all  natural  action,  viz : 

First:  All  energetic  action,  whether  classed  as  celestial, 
mechanical,  thermal,  chemical,  electrical,  biological  or  sociological, 
operates  under  the  same  general  principles  in  action.  For  in  all 
these  diverse  forms,  in  so  far  as  anything  exact  may  be  said 
about  their  structure,  energy  consists  in  the  subdivision  and 
organization,  into  specified  relationships  of  motion  and  arrange- 
ment, of  a  mass  of  material.  Yet  this  material,  of  itself,  pos- 
sesses none  of  the  characteristics  peculiar  to  the  ivhole.  The 
nature  of  a  celestial  system  is  not  determined  by  the  peculiarities 
of  its  planets,  but  by  the  peculiarities  of  their  orbital  relation- 
ships. That  of  a  machine  does  not  depend  upon  those  of  its 
component  parts  (so  long  as  they  come  up  to  certain  minimum 
requirements),  but  upon  the  way  in  which  the  engineer  has  put 
them  together.  The  features  of  a  chemical  compound  have 
nothing  to  do  with  those  of  its  component  elements.  Gases  can 
be  combined  to  produce  a  solid,  and  solids  to  produce  a  gas,  and 
vastly  greater  contrasts  between  raw  material  and  result  con- 
stantly appear  in  the  chemical  laboratory.  Our  deadliest  poisons 
and  best  foods  are  both  but  carbon,  hydrogen,  nitrogen  and  oxy- 
gen, differently  arranged.  All  are  explained  as  being  different 
relationships,  within  the  molecule,  of  elementary  atoms  which 
are  alike  for  all  known  chemical  substances. 

In  the  most  varied  physical  aspects  of  electrical  action  it  is 
the  same.  No  one  has  gone  further  than  Prof.  J.  J.  Thomson 
in  the  reduction  of  all  phenomena  to  mere  variations  in  relation- 
ship between  elementary  components  possessing  only  elementary 
characteristics;  yet  he  works  chiefly  with  electrical  concepts. 
Similarly,  the  most  intricate  variety  of  vegetable  forms  of  life 
is  shown  by  botanists  to  be  but  a  variety  of  arrangements  of  a 
substantially  uniform  vegetable  cell,  which  is  specialized  about  as 
much  to  develop  root,  stalk,  leaf  or  flower  in  any  one  plant  as  it 


TRANSFORMATIONS  AND  CONSERVATIONS    229 

is  to  embody  the  vast  differences  between  one  plant  and  another. 
And  the  same  is  true,  to  skip  briefly  to  the  other  extreme  of 
natural  action,  of  the  energetic  action  of  men.  One  method  of 
organization  will  make  of  an  army  a  panic  stricken  mob ;  another 
constitutes  it  an  invincible  foe.  One  plan  of  organization  within 
a  factory  leads  to  chronic  bankruptcy ;  another  to  opulent  profits. 
Anthropologists  tell  us  that  the  peasants  of  modern  France,  which 
leads  the  world  in  science  and  art,  are  the  exact  copies,  in  cranial 
development,  of  their  ancestors  of  eighty  thousand  years  ago. 
But  eighty  thousand  years  ago  ideas  of  political  and  economic 
organization  were  exceedingly  crude. 

The  one  necessary  lesson  for  the  clarifying  of  future  progress 
is  that  each  form  of  energy  is  defined,  fitly  for  scientific  dis- 
cussion, only  when  we  consider  activities  between  its  component 
units,  carefully  excluding  all  action  which  may  occur  within  any 
"unit."  Otherwise  is  confusion  and  no  progress.  The  "unit" 
may  be  an  electron,  or  a  molecule,  or  a  solar  system,  or  a  pro- 
toplasmic cell  (in  biological  energy),  or  a  man,  as  in  sociological 
energy.  Or,  in  the  case  of  that  international  energetic  action 
and  reaction  which  has  arisen  with  the  steamship,  the  cable  and 
the  wireless,  the  unit  of  mass  may  be  an  entire  nation.  The 
true  energy,  existing  between  the  units,  may  trade  energy  with 
its  component  units,  or  with  external  systems,  it  is  true;  but 
unless  we  exclude  these  sources  and  destinations  from  the  dis- 
cussion we  are  talking  of  two  or  three  things  at  once.  Confusion 
is  inevitable. 

Nowhere  is  this  need  of  definition  and  clarity  greater  than  in 
discussing  the  sociological  energy  existent  between  (not  within) 
individual  men.  When  we  assemble  a  hundred  metallic  parts 
into  a  machine  we  regard  the  relationships  between  the  parts  as  a 
special  form  of  energy — mechanical — and  as  a  fit  subject  for  a 
special  science,  mechanics.  We  leave  all  metallurgical  questions 
lying  within  each  piece  to  other  books  and  men.  Electricity, 
again,  we  define  as  a  relationship  between  electrons,  positive  and 
negative.  But  when  we  get  a  few  millions  of  electrons  cemented 
into  a  number  of  molecules,  we  call  the  relationship  between 
the  molecules  a  new  and  distinct  form  of  energy — chemical. 
When  we  get  a  billion  molecules  organized  into  a  protoplasmic 
cell  or  so,  we  call  this  relationship  between  the  different  chemicals 
still  another  form  of  energy — biological.  When  we  get  a  million 


230  ENERGY 

protoplasmic  cells  arranged  in  organic  form,  and  the  organs  spe- 
cialized and  federated  into  a  sentient,  reproductive  animal,  we 
regard  the  relationship  between  the  cells  and  organs  (which  are 
all  alike,  yet  all  different)  as  still  another  distinctive  form  of 
energy — that  of  the  human  individual. 

But  when  we  get  a  hundred  million  men  and  women  organ- 
ized into  special  sexes,  ages,  trades  and  professions,  and  these 
federated  and  refederated  into  a  modern  State,  we  decline  to 
admit  that  there  arises  therein  a  new  and  distinct  form  of  life 
and  energy — the  sociological — between  individuals.  We  insist 
upon  dragging  into  the  question  at  every  point  the  human  nature 
within  each  unit — each  different,  it  is  true,  yet  averaging  as  like 
as  any  million  molecules.  We  decline  to  see  that  human  institu- 
tions may  themselves  have  an  organic  life,  growth,  reproduction 
and  death — as  independent  of  the  myriad  of  individual  lives, 
growths,  passions  and  deaths  within  them,  as  the  history  of  our 
solar  system  as  a  whole  is  independent  of  the  internal  natures 
of  its  component  parts ;  which  last  are  much  more  diverse  than 
are  different  human  natures.  We  fail  to  see  that  the  written 
history  of  mankind  records  the  growth,  not  of  individual  man — 
for  evolutionary  science  declares  this  growth  to  have  been  com- 
pleted before  the  dawn  of  history — but  of  this  institutional  organ- 
ism of  human  relationships,  an  organism  as  distinct  from  indi- 
vidual man  as  the  latter  is  distinct  from  his  own  component 
protoplasmic  cells.  That  is  why  we  fail,  at  present,  of  a  con- 
sistent and  satisfactory  science  of  sociology:  we  have  not  yet 
taken  it  up  as  a  department  of  universal  energetics. 

With  our  civilization  now  approaching  a  feverish  paradox  of 
farce  and  tragedy,  with  stupendous  rates  of  production  and 
transportation  of  the  means  for  life  rising  in  rivalry  with  stupe- 
fying rates  of  poverty,  suicide,  insanity  and  crime;  with  our 
cost  of  living  rising  while  labor-saving  aids  multiply ;  with  our 
system  of  exchange  left  as  religiously  to  the  care  of  chaotic  antag- 
onism of  interests  and  duplication  of  effort  as  our  systems  of 
production  have  been  subjected  to  the  last  refinement  of  coopera- 
tive organization — with  all  these  phenomena  becoming  the  char- 
acteristic ones  of  our  world-civilization,  the  sociological  doctors 
disagree,  both  as  to  diagnosis  and  remedy,  more  and  more  hope- 
lessly. It  is  high  time  that  the  said  doctors  were  sent  back  to 
school,  and  there  impregnated  with  a  vigorous  concept  of  the 


TRANSFORMATIONS  AND  CONSERVATIONS    231 

general  principles  of  all  energetic  action.  These  apply  most 
effectively  to  every  other  problem  in  a  pretty  wide  and  intricate 
universe.  They  will  solve  our  sociological  problems.  They 
should  be  made  the  fundament  of  every  college-course,  whether 
aimed  at  pure  science  or  at  pedagogy,  at  engineering,  medicine, 
law,  the  ministry,  journalism  or  statecraft,  as  the  ultimate  goal. 

Secondly:  All  energetic  action  consists  in  a  swing  of  one  of 
the  two  great  energetic  factors — number  of  correlated  parts,  on 
the  one  hand,  or  intensity  of  relationship,  on  the  other — on  either 
side  of  a  central,  or  mean  energetic,  condition.  In  no  case  may 
any  of  the  factors  ever  reach  zero ;  none  may  ever  reach  infinity. 
And  this  swing  occurs  under  the  guidance  and  propulsion,  as 
also  against  the  resistance,  of  two  great  gravitational  tendencies, 
one  toward  the  consolidation  and  the  other  toward  the  disgrega- 
tion  of  the  component  material. 

These  two  gravitational  tendencies  are  never  directly  opposed. 
Each  is  disposed  laterally  or  transversely  to  the  other.  Within 
certain  limits,  each  may  act  independently  of  the  other.  Between 
the  two,  therefore,  may  occur  the  greatest  variety  of  lateral  per- 
turbations of  the  general  swing  from  one  extreme  to  the  other. 
The  pendulum  of  energetic  conditions  is  not  confined  to  a  single 
plane,  but  is  capable  of  the  utmost  variety  of  cyclical  gyrations — - 
never  confining  itself  to  any  regular  geometric  path,  and  seem- 
ingly intricate  in  its  motions  beyond  comprehension — yet  always 
guided  in  an  eternal  stability  of  equilibrium.  In  whatever  direction 
departure  may  be  made,  the  departure  itself  begets  the  disposition 
to  return.  Whether  the  swing  be  confined  within  the  critical  limits, 
concerned  merely  in  an  exchange  of  one  dimension  of  energy  for 
the  other,  or  whether  it  trespass  beyond,  begetting  energy- 
transformation,  the  equilibrium  continues  ever  stable.  There  is 
no  activity  in  nature,  inanimate  or  animate,  which  does  not  vary 
stably  about  a  mean  central  condition,  from  which  it  never  can  be 
driven,  by  vagary  of  circumstance,  more  than  a  finite  distance, 
or  against  less  than  a  proportionally  increasing  resistance  which 
must  eventually  reverse  the  process  into  a  return. 

Thirdly:  This  central  or  mean  energetic  condition,  which 
neither  requires  nor  is  capable  of  any  fixed  support  from  any 
rigid  base,  but  hangs  in  mid-space  like  the  sun  in  the  heavens, 
contains  always  an  immeasurable  amount  of  latent  and  invisible 
"tangential"  energy,  into  which  and  out  of  which  the  perceptible 


232  ENERGY 

or  "sensible"  funds  of  radial  energy  pass  in  indefinite  amount. 
There  is  no  department  of  natural  action  which  has  proven  so 
deceptive  to  the  engineer  as  that  of  latent  energy.  The  instances 
are  legion.  The  most  dramatic  illustrations  can  be  drawn,  as 
usual,  from  social  energetic  systems,  wherein  the  politicians,  if 
not  the  statesmen,  are  continually  deceived  as  to  the  energetic 
possibilities  latent  within  a  people,  and  as  to  when  they  are  likely 
to  burst  forth.  In  1776  Great  Britain  was  nonplussed  by  the 
indomitable  resistance  of  a  few  ragged  American  farmers.  In 
1795,  when  France  at  home  was  a  howling  mob,  unable  to  enforce 
order  or  supply  bread  on  the  streets  of  Paris,  her  armies  scat- 
tered the  combined  forces  of  England,  Prussia,  Austria  and 
Italy.  In  America  again,  in  1864,  the  South  and  the  copperheads 
could  not  understand  whence  came  the  unending  resistance  of 
the  North — though  Gladstone's  insight  told  him  that  the  South, 
with  England  behind  it,  was  "righting  the  law  of  gravitation." 

To-day  the  same  is  true.  The  engineers  are  as  blind  as  the 
politicians,  in  their  failure  to  comprehend  the  enormous  latent 
possibilities  for  productive  energetic  action  which  lie  in  the 
armies  of  workmen  of  which  they  must  always  be  the  officers. 
If  only  these  armies  be  once  organized  for  a  single  harmonious 
end,  their  energy,  according  to  the  laws  of  energetics,  must 
increase  more  rapidly  than  the  first  power,  and  possibly  as  fast 
as  the  second  power,  of  the  numbers  involved.  That  single  end 
— in  order  to  develop  this  best  rate  of  increase — must  be  neither 
mere  volume  of  material  output  nor  accentuation  of  dividends, 
although  to-day  these  are  the  sole  aim  of  the  engineer-adminis- 
trator. It  must  be,  instead,  the  welfare  of  the  consumer,  for 
whose  support  alone  exists  the  entire  economic  system.  The 
present  division  of  society,  in  public  opinion,  into  several  classes, 
such  as  laborer,  employer,  capitalist,  etc.,  of  which  the  consumer 
appears  as  merely  one,  must  cease.  In  economic  democracy, 
none  of  these  possesses  any  rights  whatever  except  as  a  con- 
sumer. In  primitive  nature  the  law  of  hunger  drove  the  arm 
of  toil  inexorably.  In  intricate  societies  it  must  be  the  same. 
Yet  in  modern  economics  the  law  of  supply  and  demand  operates 
but  remotely,  obscurely  and  indirectly.  Like  a  river  under- 
ground, or  one  obstructed  by  dams  and  dikes  and  diverted  into 
artificial  channels,  it  flows  and  feeds.  But  that  is  all.  It  is  not 
free  and  it  does  not  control. 


TRANSFORMATIONS  AND  CONSERVATIONS    233 

It  is  just  as  natural  that  a  growing  population  should  be 
increasingly  self-supporting  as  it  is  that  a  locomotive  or  steam- 
ship should  be  increasingly  able  to  haul  more  coal  than  it  burns. 
But  if  we  should  design  our  locomotives  for  the  much  more 
spectacular  purpose  of  maintaining  a  pyrotechnic  display  of 
sparks  by  night  and  a  salvo  of  steam-fountains  by  day,  rather 
than  for  coal-hauling,  we  might  easily  find  that  they  could  not  < 
haul  coal  enough  even  to  supply  their  own  consumption.  The 
more  such  locomotives  we  built  the  poorer  we  should  be. 

Yet  our  present  method  of  constructing  economic  systems — 
or  rather,  of  tolerating  unchanged  those  which  we  have  inherited 
from  an  ignorant  past — is  quite  as  this.  The  real  and  sole  reason 
for  the  existence  of  an  economic  system  at  all — the  transfer  of 
food  from  the  soil  to  the  mouth — has  been  almost  totally  for- 
gotten. All  now  centers  upon  considerations  of  immediate  profit 
from  intermediate  exchange,  multiplied  for  the  purpose.  Our 
economic  system  operates  primarily  for  the  purposes  of  profitable 
antagonism,  empty  display  and  concealed  gains — with  huge  inci- 
dental waste.  The  consumer  is  supposed  to  have  all  proper  control 
over  those  activities,  which  his  money  alone  hires  into  being, 
when,  at  the  bargain-counter,  he  chooses  between  two  or  more 
parallel  lines  of  those  activities ;  which  lines  are  generally  as  like 
as  two  peas.  The  fact  that  he  alone  pays  all  the  bills,  for  raw 
material,  labor,  superintendence,  finance,  dividends,  profits,  and 
finally  the  cost  of  persuading  himself  to  buy  what  he  actively 
desires  or  urgently  needs,  and  that  therefore  his  right  to  control 
goes  as  far  as  does  his  dollar,  has  been  quite  forgotten. 

Instead,  intermediate  means  are  confused,  as  guides  in  organ- 
ization, with  these  sole  ends — the  sustenance  and  profit  of  the 
consumer.  The  costly  strife  for  private  profit  at  intermediate 
points,  the  purposeless  paying  of  dividends  for  the  purpose  of 
enticing  into  reinvestment  a  remnant  of  those  same  dividends, 
under  the  guise  of  timid,  though  very  "willin*  "  capital,  and  for 
the  support  of  a  pyrotechnic  spectacle  of  luxury  which  forms  no 
essential  part  of  production  and  distribution,  has  overshadowed 
the  sustenance  of  the  real  life  of  the  community  as  the  sole  object 
of  all  business.  Incidentally  thereto  has  arisen,  with  the  increasing 
complexity  of  invention  and  the  arts,  a  rapidly  increasing  intri- 
cacy and  intensity  of  confusion  between  all  industrial  organiza- 
tions, which  no  business-man  would  tolerate  for  a  moment 


234  ENERGY 

within  any  one  of  them  which  he  controlled.  With  the  com- 
mercial and  technical  press  daily  calling  with  greater  vehemence 
for  refinement  of  factory-organization,  for  better  efficiency  of 
result,  the  productive  organization  of  the  community  as  a  whole 
is  daily  presenting  a  more  hopeless  chaos  of  cross-purposes, 
antagonism  of  interest  and  duplication  of  effort,  resulting  in  a 
rapidly  decreasing  efficiency  of  result — the  feeding  of  the  con- 
sumer. With  labor-saving  and  luxury-creating  invention  advanc- 
ing at  a  rate  never  before  known  in  history,  the  cost,  difficulty, 
uncertainty  and  dissatisfaction  of  living  are  daily  upon  the 
increase.* 

It  is  the  organization,  therefore,  of  all  industrial  enterprises 

*Should  the  reader  be  interested  in  following  more  in  detail  this  grow- 
ing inefficiency  of  our  system  of  industry  and  exchange,  he  will  find  the 
same  analyzed  and  displayed,  for  the  half-century  of  American  progress 
from  1850  to  1900,  in  the  writer's  "Cost  of  Competition."  He  will  find 
there  the  proofs  that,  whereas  in  1850  the  efficiency  of  organization — quite 
aside  from  any  question  of  efficiency  of  individual  effort — was  such  that 
seventy  per  cent,  of  the  effort  exerted  was  transformed  into  useful  result, 
while  thirty  per  cent,  was  dissipated  in  commercial  impact  and  friction, 
over  questions  of  price  and  ownership  which  are  of  no  interest  whatever 
to  the  consumer,  so,  in  contrast,  in  1900  these  figures  had  become  almost 
exactly  reversed.  Of  the  effort  now  being  expended  within  our  commercial 
system  less  than  one-third  results  in  useful  product;  and  that  small  frac- 
tion suffices  to  produce  all  which  we  now  consume.  Of  this  same  effort 
actually  expended  fully  seventy  per  cent,  is  being  currently  lost,  in  impact 
and  friction  due  to  sheer  lack  of  intelligent  organization  between  factory 
and  factory. 

The  daily  progress  of  our  times,  as  revealed  in  the  weekly  reviews,  can- 
not be  understood  except  from  these  facts  as  a  basis:  that  with  the  issue 
of  every  new  patent,  with  the  landing  of  each  new  immigrant,  life  grows 
more  complex.  The  demand  for  extension  of  organization  grows  more 
urgent.  It  is  neither  the  poverty  nor  the  criminality  of  the  immigrant 
that  is  the  trouble.  Both  prove,  upon  rigid  investigation,  to  be  imaginary. 
The  immigrant  is  a  raw  material  of  the  greatest  potential  value.  But, 
dumped  upon  a  land  wherein  economic  organization  is  proceeding  at  a  rate 
far  below  that  required  by  natural  law — far  below  all  other  rates  of 
progress— its  effect  is  that  of  a  ton  of  coal  dumped  on  a  furnace-fire  need- 
ing^ a  hundred-weiprht.  Valuable  as  are  both  invention  and  immigration, 
it  is  their  combination  which  is  now  forcing  the  country  into  economic 
instability.  We  are  much  further  from  assimilating  properly,  by  industrial 
combination,  the  currentjnflux  of  invention  than  we  are  from  that  of  immi- 
gration. Beneficial  as  is  the  industrial  combination  of  tangible  proper- 
ties into  greater  unitv — for  it  is  the  major  source  of  our  wonderful  eco- 
nomic progress  in  the  recent  past— it  is  now  proceeding  at  a  rate  far  below 
that  requisite  for  the  control  of  social  intensities  below  the  critical  point, 
for  the  preservation  of  stability  of  ecmilibrium  and  for  the  prevention  of 
explosive  energy-transformations  as  far-reachine  and  destructive  as  those 
involved  jn  the  abolition  of  shverv.  The  only  solution  lies  in  the  far  more 
rapid  unification  of  all  industrial  properties— at  a  rate  nearly  proportional 
to  the  square  of  the  population. 


TRANSFORMATIONS  AND  CONSERVATIONS    235 

into  a  single  whole,  along  exactly  the  same  lines  as  those  now 
enforced  by  all  business-men  in  the  organization  of  individual  \ 
men  within  these  enterprises,  which  alone  can  develop  within  the 
community  its  quantity-factor  of  social  energy  into  commen- 
suracy  with  its  growing  needs,  which  alone  can  expand  its  pro- 
ductive capacity  more  than  proportionally  to  the  first  power  of 
its  population.  It  is  the  engineers  of  the  community,  more  than 
any  other  one  class,  who  must  perform  this  task.  It  is  they, 
above  all  others,  who  are  equipped  for  understanding  what  they 
are  about  as  they  do  this  thing. 

If  this  task  be  not  undertaken  the  critical  limits  of  accumu- 
lated intensity  will  soon  be  passed.  Intelligence  will  then  have 
become  impotent.  Forces  will  have  been  released  which  must 
then  have  their  brutal  sway  uncontrollably,  until  stability  be 
regained  through  exhaustion.  Economic  equilibrium,  already 
wavering,  will  have  become  grossly  unstable.  Explosion  must 
ensue.  It  is  not  that  labor  will  strike  successfully  against 
capital.  There  were  as  many  chances  for  that  three  centuries 
ago  as  now.  It  is  that  the  hundred  million  consumers  will 
strike  against  the  absurd  strife  and  confusion  now  prevailing, 
not  only  between  labor  and  capital,  but  between  capital  and 
capital,  leading  to  such  universal  impact  and  friction  that  ineffi- 
ciency of  result  is  growing  upon  us  apace.  They  will  rise  and 
overthrow  in  its  entirety  a  system  which,  in  the  twentieth  cen- 
tury, can  still  find  no  better  foundation  and  guide  than  universal 
antagonism  of  interest  between  man  and  man,  between  enterprise 
and  enterprise — an  antagonism  to-day  artificially  stimulated  to 
the  last  degree,  in  a  vain  endeavor  to  rouse  it  to  a  task  for 
which  it  is  inherently  and  inevitably  impotent.  No  imaginable 
expansion  of  intensity  of  economic  energy  can  ever  meet  a  need 
for  its  greater  extensity.  Its  only  effect  can  be  to  exaggerate, 
as  it  delays,  the  vigor  of  the  inevitable  reaction. 

We  shall  be  surprised,  in  looking  back  upon  this  crisis  when 
it  is  passed,  to  see  how  largely  its  incurrence  has  been  due  to 
the  fact  that  the  business-men,  factory-superintendents  and 
mechanical  engineers  of  a  mechanical  people  do  not  understand 
the  true  nature  of  mechanical  and  allied  energies. 

Conservations.  Throughout  all  this  wonderful  intricacy  of 
energy-transformation,  between  work,  heat,  chemical  and  elec- 
trical action,  light,  radioactivity,  vegetation,  animal  life,  the 


236  ENERGY 

activities  of  the  body  politic  and  economic,  and  the  latent  strength 
of  that  vast  whirl  of  international  human  solidarity — the  accumu- 
lation which  constitutes,  in  the  eyes  of  mankind,  the  highest  aim 
of  all  these  combined — throughout  this  whole,  for  all  time,  run 
the  three  great  principles  of  eternal  conservation,  the  first  scien- 
tific statements  of  immortality.  All  that  we  can  see  or  know  as  a 
Thing,  throughout  all  this  limitless  intricacy  of  things,  proves, 
upon  examination,  to  be  a  mere  temporary  form.  It  is  a  form 
of  relationship,  between  component  portions  none  of  which  pos- 
sesses the  attributes  or  abilities  of  the  Thing  itself.  These  attri- 
butes and  abilities  are  the  property  of  the  form  of  relationship 
only.  As  this  form  is  created,  either  from  formless  dissociate 
dust  by  its  congregation  into  organized,  interacting  propinquity, 
or  from  senseless  solidity  by  its  comminution  and  disgregation 
into  energetic  sensitiveness,  these  attributes  and  abilities  come 
into  existence.  As  the  form  of  relationship  changes,  so  do  the 
attributes  and  abilities.  As  the  form  of  relationship  melts  again 
into  formless  dissociation,  or  solidifies  into  passive  stolidity,  the 
attributes  and  abilities  disappear. 

Creation,  birth,  life  and  death  are  of  form  only.  The  one 
life  of  the  universe  continues  unceasingly  and  unvaryingly.  It 
is  the  reality  alone,  not  the  form,  of  the  universe  which  is 
eternal.  It  is  that,  too,  which  is  imperceptible;  which  possesses 
neither  attributes,  character  nor  individuality.  It  is  not  alone 
that  the  human  senses  take  no  cognisance  of  aught  but  mere 
form.  They  take  cognisance  only  of  change  in  form.  If  it 
were  not  for  the  birth,  life  and  death  of  many  millions  of  ether- 
forms  before  the  eye  each  second,  we  should  see  no  light.  Were 
it  not  for  ceaseless  alteration  of  air-pressure  upon  our  ear-drums 
we  should  live  in  blank  silence.  Were  it  not  for  ceaseless  chem-' 
ical  metabolism  of  carbon,  hydrogen,  oxygen  and  nitrogen — 
themselves  unchanging — in  vegetable  life,  we  should  live  in  a 
desert;  though  sun,  moon  and  earth  still  circled,  we  should  have 
no  seasons;  winter  would  mean  a  cold  bare  rock  and  summer  a 
hot  one,  equally  bare.  Were  it  not  for  the  ceaseless  birth,  change 
and  death  of  countless  cells  comprising  our  own  bodies,  so  that 
we  possess  none  of  the  flesh  which  we  inhabited  a  few  months 
ago,  we  should  know  no  individual  life  and  growth.  Were  it 
not  for  the  ceaseless  procession  of  new-born  babies  into  the 
world,  of  children  shooting  up  into  the  bloom  of  adolescence,  of 


TRANSFORMATIONS  AND  CONSERVATIONS    237 

active  lives  grown  seamed  and  scarred  and  feeble  from  the  buffets 
of  fate,  of  old  people  laid  lovingly  away  to  rest — there  could  be 
no  progress  and  history  of  the  human  race. 

What  folly,  then,  to  speak  of  inducing  progress!  Progress 
occurs  because  it  cannot  help  itself.  Behind  it  is  the  energy  of 
countless  eons,  non-creatable  and  indestructible.  As  well  invite 
the  earth  to  move  more  rapidly  about  the  sun,  as  well  invite  the 
vine  to  grow  more  rapidly  than  its  natural  rate  for  soil  and  sun 
provided,  as  to  attempt  to  invite  or  force — or  quell  or  retard,  for 
that  matter — human  progress.  Not  only  the  energy,  but  also 
the  forward  motion,  of  the  race  is  indestructible  and  non- 
creatable.  As  mankind  entire  is  but  a  bit  of  microscopic  growth 
on  the  surface  of  a  tiny  mass-portion  whirling  in  space,  the  last 
and  most  delicate  fruit  of  ages  of  upward  struggle  on  the  part 
of  trilobite  and  dinosaur,  so  are  his  energies  but  the  latest  form 
and  conservation  of  the  measureless  energy-fund  of  the  universe, 
whirled  back  and  forth  across  abysmal  space  with  inconceivable 
speed,  but  incapable  of  being-  lost  or  retarded,  or  increased  or 
accelerated,  by  the  slightest  iota. 

The  fundamental  principles  of  energetic  conservation  upon 
which  these  conclusions  rest  are  these — stated  in  terms  of 
mechanical  energy,  as  the  simplest  and  most  familiar  form,  but 
interpretable  in  terms  of  any  and  all  known  energy-forms. 

I.  The  FIRST  Law  of  Energetics:   the  Conservation  of 
MASS.     Mass  is  quantity  of  matter.     It   exists   eternally.     It 
undergoes    local    and    temporary    aggregation    or    disgregation, 
ceaselessly ;  but  it  is  never  destroyed  nor  created. 

II.  The  SECOND   Law  of  Energetics:    the  Conserva- 
tion of  ENERGY.     Energy  is  the  space-and-motion  relationship 
between    separate   portions   of    mass.      It    exists    eternally.      It 
undergoes   local   and   temporary    accumulation,    dissipation    and 
transformation,  ceaselessly;  but  it  is  never  destroyed  nor  created. 

III.  The  THIRD  Law  of  Energetics:   the  Conservation 
of  INTENSITY  or  AVAILABILITY  of  Energy.     Intensity 
is  the  degree  of  spacial  propinquity  and  of  linear  motion  between 
the  separate  mass-portions.     It  exists   eternally.     It  undergoes 
local  and  temporary  concentration  or  diffusion,  ceaselessly;  but 
it  is  never  destroyed  nor  created. 


238  ENERGY 

IV.  The  FOURTH  Law  of  Energetics:  the  Conserva- 
tion of  EXTENSITY  of  Energy.  Extensity  is  the  extent  of 
relationship,  or  mass-pairing,  between  the  interacting  portions  of 
mass.  It  is  what  embodies  the  intensity  of  energy,  to  give  the 
latter  a  habitation  and  a  name  with  which  to  do  its  work.  It 
exists  eternally.  It  undergoes  local  and  temporary  accentuation 
or  disguise,  through  the  aggregation  or  disgregation  of  mass, 
ceaselessly ;  but  its  sum  total  is  never  created  nor  destroyed. 


THE  END. 


INDEX 

PAGE 

Absolute  zero    79 

Adiabatic    130 

Angle  of  Incidence    24,  35 

Apastron    22 

Basic    Energetic    Processes     129,  156,  163 

Carnot  cycle  166 

Clausius  136 

Combative  energy  55 

Conservations  208,  235 

Conservation  of  Energy  18,  137,  237 

of  Mass  :8,  237 

Contact  27,  114 

Copernican  philosophy  223 

Critical  energetic  conditions  61,  66 

limits  of  intensity C6,  218 

Cycle 158 

Cycle-efficiency  169,  171 

9 

Density 126 

Dimensions  of  energy  48,  78,  162,  214 

Dualism    in   energetics    48,  78,  162,  213 

Eccentricity    25,  34,  47 

Elasticity    90,  122,  143 

Energetics,   Dualism  in    213 

Elements  of    17,30,46,74,76,78,163,237 

Energetic  Action.     Unity  of  all    225 

Cycle   158 

Equilibrium         84,  195,  217,  231 

Form.     Interchangeably   of 209,  211 

Gravitation       195 

Universe.     Fundament  of  222 

Energy-fund    32,  35 

Energy-transfer    112 

Energy-transformation     217 

Cause   of    79,  87 

Equilibrium    in    203 

Fundamental    equation    for    18 


INDEX—  Continued 

PAGE 

Entropy        100,  131,  136,  142,  151,  156,  157 

Athletic      155 

Combative  or  military  55 

Entropy-temperature   diagram    95,  100,  136,  190 

Equilibrium.    Energetic   84,  203,  219 

Between  interchangeable  forms  of  energy 203,219 

Intramolecular    152 

Thermal    182 

Extensity   of   energy   48,  51,  78,  162 

Extreme  energetic  conditions   61 

Factors  of  energy    48,  78,  162,  214 

First  law  of  energetics  31,  237 

Fourth  law  of  energetics  238 

Free  motion   18,  30 

Fundament  of  the  energetic  universe  222 

Gravitation.    Energetic    81,  195,  204 

Mechanical 13,  67,  196 

Thermal    195 

Heat    89,  144 

Heat-transfer    129,  134,  143 

Inelasticity    90 

Intensity  of  energy   48,  78,  162,  195 

Gravitation  of  204 

Rejuvenation  of   203 

Summation  of  205 

Interchangeability  of  energetic  form 211 

Irregular  cycles   175,  178 

Isomorphic    98 

Isothermal   99 

Kepler's   laws    29 

Kepler's  and  Newton's  laws  combined 30 

Kinetic   energy    15 

Labority    135,  143,  149,  156 

Latent  heat   99,  169 

Lower  critical  intensity   66 

Mass,  Conservation  of  18,  237 

Mass-pairing    48,  51,  139,  154,  173 

Mean  energetic  condition   32,  79,  81,  82,  190,  222,  231 

distance    33 


INDEX— Continued 

PAGE 

Mechanical    energy    9,  30,  78 

Basic  processes  of   163 

Mechanical  universe 30 

Metamorphic    98 

Metathermal 98,  129 

Military  energy  55 

Natural   action    30 

Newton's    laws    13,  28 

Newton's  and  Kepler's  laws  combined 30 

Parabola    25,  34,  76,  222 

Periastron    22 

Permanent  energy   74,  93,  202 

Potential    energy    14,  41 

Pressure.     Mechanical  concept  of 106,  145 

Primary  and  secondary  energy- forms    175 

Propinquity    41 

Radial  energy    35,  40,  143 

intensity   65 

Rej  uvenation  of  energy  203 

Relativity    59,  228,  236 

Reversed   cycles    175 

Reversibility    136,  172 

Second  law  of  energetics  ". 31,  237 

of    thermodynamics    205 

Sensible  heat   99 

Stability  of  energetic  equilibrium    84,  203,  219 

Subpermanent  and  superpermanent  energies   74 

Summation  of  energetic  intensities  205 

Tangential    energy    35,  41,  143 

Temperature    95,  141,  145,  151,  156 

Thermal   conduction    129 

Diagram 95,  100,  190 

gravitation    195 

energetics.     Basic  processes  of 129,  156 

equilibrium  182,  201 

Thermogy   134,  143,  148,  156 

Third  law  of  energetics 237 

Transformation  of  energy    79,  87,  208 

Vibratory    energies    18: 

Volume.     Mechanical  concept  of 106,  145 

Water-wheel   cycle   160 

Wire-drawing    130 

Work-performance    129,    135,  143,  149,  156 

Zero.     Absolute    79,  98 


^ 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
This  book  is  DUE  on  the  last  date  stamped  below. 


OCT20  19*7 


REC'D  LD 

JUN  1  2  1961 

INTERI.IBRARY  LOAN 

41983 
UNIV.  OF  CALIF   r' 

LD  21-100m-12,'46(A2012sl6)4120 


' 


/ 


YD  02706 


.: 


A/A'3   '     /-£' 


